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Mathematics and History of Complex Variables

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Solo Hermelin
Solo Hermelin

Mathematics of complex variables, plus history. This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level. Please send comments and suggestions to solo.hermelin@gmail.com, thanks! For more presentations, please visit my website at http://www.solohermelin.com

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1 Complex Variables SOLO HERMELIN Updated: 11.05.07 28.10.12 http://www.solohermelin.com
2 SOLO Complex Variables Table of Contents Set of Numbers – Examples Fundamentals Operations with Complex Numbers z = x + i y Axiomatic Foundations of the Complex Number System ( ) R∈= babaz ,, History of Complex Numbers Derivatives Cauchy-Riemann Equations Harmonic Functions Orthogonal Families Singular Points Complex Line Integrals Simply and Multiply Connected Regions Green’s Theorem in the Plane Consequences of Green’s Theorem in the Plane Cauchy’s Theorem Cauchy-Goursat Theorem Consequences of Cauchy-Goursat Theorem
SOLO Complex Variables Table of Contents (continue - 1) Cauchy’s Integral Formulas and Related Theorems Cauchy’s Integral Formulas Cauchy’s Integral Formulas for the n Derivative of a Function Morera’s Theorem (the converse of Cauchy’s theorem) Cauchy’s Inequality Liouville’s Theorem Foundamental Theorem of Algebra Gauss’ Mean Value Theorem Maximum Modulus Theorem Minimum Modulus Theorem Poisson’s Integral Formulas for a Circle Poisson’s Integral Formulas for a Half Plane
4 SOLO Complex Variables Table of Contents (continue - 2) Theorems of Convergence of Sequences and Series Convergence Tests Cauchy Root Test D’Alembert or Cauchy Ratio Test Maclaurin or Euler Integral Test Kummer’s Test Raabe’s Test Gauss’ s Test Infinite Series, Taylor’s and Laurent Series Infinite Series of Functions Absolute Convergence of Series of Functions Uniformly Convergence of Sequences and Series Weierstrass M (Majorant) Test Abel’s Test Uniformly Convergent Series of Analytic Functions Taylor’s Series Laurent’s Series (1843)
5 SOLO Complex Variables Table of Contents (continue - 3) 5 The Argument Theorem Rouché’s Theorem Foundamental Theorem of Algebra (using Rouché’s Theorem) Zeros of Holomorphic Functions Theorem: f(z) Analytic and Nonzero → ln|f(z)| Harmonic Polynomial Theorem Jensen’s Formula Poisson-Jensen’s Formula for a Disk
6 SOLO Complex Variables Table of Contents (continue - 4) Calculation of the Residues The Residue Theorem, Evaluations of Integral and Series The Residue Theorem Evaluation of Integrals Jordan’s Lemma Integral of the Type Bromwwich-Wagner Integral of the Type ,F (sin θ, cos θ) is a rational function of sin θ and cos θ ( )∫ π θθθ 2 0 cos,sin dF Definite Integrals of the Type .( )∫ +∞ ∞− xdxF Cauchy’s Principal Value Differentiation Under Integral Sign, Leibnitz’s Rule Summation of Series Infinite Products The Mittag-Leffler and Weierstrass , Hadamard Theorems The Weierstrass Factorization Theorem The Hadamard Factorization Theorem Mittag-Leffler’s Expansion Theorem Analytic Continuation Conformal Mapping
7 SOLO Complex Variables Douglas N. Arnold Gamma Function Bernoulli Numbers Fourier Transform Laplace Transform Z Transform Mellin Transform Hilbert Transform Zeta Function Table of Contents (continue - 5) Applications of Complex Analysis References
8 SOLO Algebra Set of Numbers – Examples { }+∞<<∞−= xnumberrealaisxx ,:R Set of real numbers { }+∞<<−∞−=+== yxiyixznumbercomplexaiszzC ,,1,,: Set of complex numbers { } ,3,2,1,0,1,2,3,,: −−−= integeranisiiZ Set of integers { },3,2,1,0,0: integernaturalaisnnN ≥= Set of positive integers or natural numbers { }0,,,/: ≠∈== qZqpwhereqprrQ Set of rational numbers We have: CZN ⊂⊂⊂ R { }QxxIR −∈= R: Set of irrational numbers ∅== IRQIRQ  &R
9 SOLO Complex Variables Complex numbers can result by solving algebraic equations a cabb x 2 42 1 −−− = a cabb x 2 42 2 −+− = 042 >−=∆ cab a b xx 2 21 − == 042 =−=∆ cab 042 <−=∆ caby x cxbxay ++= 2 a bcaib x 2 4 2 2,1 −+− = y x dxcxbxay +++= 23 Three real roots for y = 0 One real & two complex roots for y = 0 πki ez 25 1== y x 5 2 2 π i ez = 5 2 2 2 π i ez = 5 2 3 3 π i ez = 5 2 4 4 π i ez = 11 =z  72  72  72  72  72 1. Quadratic equations 2. Cubic equations 3. Equation Examples 02 =++ cxbxa 023 =+++ dxcxbxa 015 =−x Return to Table of Contents
10 SOLO Complex Variables Fundamentals Operations with Complex Numbers z = x + i y ( ) 1 ,sincos , 2 −=    ==+ ==+ = i ArgumentModulusi partImaginaryypartRealxyix z θρθθρ θ ρ i eyixz =+= y x ρ θ Division Addition ( ) ( ) ( ) ( )dbicadicbia +++=+++ Subtraction ( ) ( ) ( ) ( )dbicadicbia −+−=+−+ Multiplication ( ) ( ) ( ) ( )cbdaidbcadbicbidaicadicbia ++−=+++=++ 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )dccdidc dacbidbca dic dic dic bia dic bia −++ −++ = − − + + = + + 22 ( ) ( ) ( ) ( ) ( ) ( ) 022 2222 ≠+ + − + + + = + + dc dc dacb i dc dbca dic bia Conjugate ( )θθρ sincos:* iyixz −=−= Absolute Value ρ==+= *22 : zzyxz θ ρ i eyixz − =−=:* θ ρ i eyixz =+=: x y ρ ρ θ θ−
11 SOLO Complex Variables Fundamentals Operations with Complex Numbers z = x + i y θ ρ i eyixz =+= y x ρ θ Polar Form of a Complex Number Multiplication Division ( )2121 212121 θθθθ ρρρρ + =⋅=⋅ iii eeezz *22 zzyxz =+==ρ ( )θθρ sincos: iyixz +=+= ( )xy /tan 1− =θ ( )2121 212121 /// θθθθ ρρρρ − == iii eeezz Euler’s Formula ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )θθ θθθθθθ θθθ θθ θ sincos !12 1 !3!1!2 1 !4!2 1 !!2!1 1 sin 123 cos 242 2 i k i k n iii e k k k k n i +=             + + −++−++−+++−= +++++= +          Leonhard Euler 1707- 1783 ( ) 1 ,sincos , 2 −=    ==+ ==+ = i ArgumentModulusi partImaginaryypartRealxyix z θρθθρ
12 SOLO Complex Variables Fundamentals Operations with Complex Numbers z = x + i y 1 , , 2 −=    == ==+ = i ArgumentModuluse partImaginaryypartRealxyix z i θρρ θ θ ρ i eyixz =+= y x ρ θ Polar Form of a Complex Number θ ρ i eyixz =+=: De Moivre Theorem ( )[ ] ( ) ( )θθρρ ρθθρ θ θ nine eiz nnin ninn sincos sincos +== =+= Roots of a Complex Number [ ] ( ) ( )[ ] 1.2.1.0 2 sin 2 cos sincos /1 /1/12/1 −=            + +      + = +== + nk n k i n k iez n nnkin  πθπθ ρ θθρρ πθ πki ez 25 1== y x 5 2 2 π i ez = 5 2 2 2 π i ez = 5 2 3 3 π i ez = 5 2 4 4 π i ez = 11 =z  72  72  72  72  72 Abraham De Moivre 1667 - 1754 Return to Table of Contents
13 SOLO Complex Variables Axiomatic Foundations of the Complex Number System ( ) R∈= babaz ,, Definition of Complex System: From those relation, for any complex numbers z1,z2,z3 ∈ C we obtain: CzzCzzCzz ∈∀∈⋅∈+ 212121 ,& Closure Law1 1221 zzzz +=+ Commutative Law of Addition2 ( ) ( ) 312321 zzzzzz ++=++ Associative Law of Addition3 1221 zzzz ⋅=⋅ Commutative Law of Multiplication4 ( ) ( ) 312321 zzzzzz ⋅⋅=⋅⋅ Associative Law of Multiplication5 ( ) 3121321 zzzzzzz ⋅+⋅=+⋅ Distributive Law6 111 00 zzz =+=+ 111 11 zzz =⋅=⋅7 0.. 11 =+∈∃∈∀ zztsCzuniqueCz zz −=18 1..0 11 =⋅∈∃∈≠∀ zztsCzuniqueCz zzz /11 1 == − 9 Equality ( ) ( ) dbcadcba ==⇔= ,,,A Sum ( ) ( ) ( ) ( )dbcadcba +++=+ ,,B Product ( ) ( ) ( )cbdadbcadcba +−=⋅ ,,, ( ) ( ) R∈= mbmambam &,,C Return to Table of Contents
14 SOLO Complex Variables History of Complex Numbers Brahmagupta (598-670) writes Khandakhadyaka (665) which solves quadratic equations and allows for the possibility of negative solutions. Brahmagupta 598 - 670 Brahmagupta also solves quadratic equations of the type a x2 + c = y2 and a x2 - c = y2 . For example he solves 8x2 + 1 = y2 obtaining the solutions (x,y) = (1,3), (6,17), (35,99), (204,577), (1189,3363), ... For the equation 11x2 + 1 = y2 Brahmagupta obtained the solutions (x,y) = (3,10), (161/5,534/5), ... He also solves 61x2 + 1 = y2 which is particularly elegant having x = 226153980, y = 1766319049 as its smallest solution.
15 SOLO Complex Variables History of Complex Numbers Abraham bar Hiyya Ha-Nasi ‫הנשיא‬ ‫חייא‬ ‫בר‬ ‫אברהם‬ writes the work Hibbur ha-Meshihah ve-ha-Tishboret ‫והתשבורת‬ ‫המשיחה‬ ‫חבור‬ , translated in 1145 into Latin as Liber embadorum, which presents the first complete solution to the quadratic equation. Abraham bar Hiyya Ha-Nasi (‫הנשיא‬ ‫חייא‬ ‫בר‬ ‫אברהם‬ Abraham son of [Rabbi] Hiyya "the Prince") (1070 - 1136?) was a Spaish Jewish Mathematician and astronomer, also known as Savasorda (from the Arabic ‫الشرطة‬ ‫صاحب‬ Sâhib ash-Shurta "Chief of the Guard"). He lived in Barcelona. Abraham bar iyya ha-NasiḤ [2] (1070 – 1136 or 1145)
16 SOLO Complex Variables History of Complex Numbers Nicolas Chuquet (1445 – 1488) Chuquet wrote an important text Triparty en la science des nombres. This is the earliest French algebra book . The Triparty en la science des nombres (1484) covers arithmetic and algebra. It was not printed however until 1880 so was of little influence. The first part deals with arithmetic and includes work on fractions, progressions, perfect numbers, proportion etc. In this work negative numbers, used as coefficients, exponents and solutions, appear for the first time. Zero is used and his rules for arithmetical operations includes zero and negative numbers. He also uses x0 = 1 for any number x. The sections on equations cover quadratic equations where he discusses two solutions.
17 SOLO Complex Variables History of Complex Numbers Girolamo Cardano 1501 - 1576 Nicolo Fontana Tartaglia 1500 - 1557 Solution of cubic equation x3 + b x2 +c x +d = 0 The first person to solve the cubic equation x3 +b x = c was Scipione del Ferro (1465 – 1526), but he told the solution only to few people, including his student Antonio Maria Fior. Nicolo Fontana Tartaglia, prompted by the rumors, manage to solve the cubic equation x3 +b x2 = -d and made no secret of his discovery. Fior challenged Tartaglia, in 1535, to a public contest, each one had to solve 30 problems proposed by the other in 40 to 50 days. Tartaglia managed to solve his problems of type x3 +m x = n in about two hours, and won the contest. News of Tartaglia victory reached Girolamo Cardan in Milan, where he was preparing to publish Practica Arithmeticae (1539). Cardan invited Tartaglia to visid him and, after much persuasion, made him to divulge his solution of the cubic equation. Tartaglia made Cartan promise to keep the secret until Tartaglia had published it himself.
18 SOLO Complex Variables History of Complex Numbers Girolamo Cardano 1501 - 1576 Nicolo Fontana Tartaglia 1500 - 1557 Solution of cubic equation x3 + b x2 +c x +d = 0 After Tartaglia showed Cardan how to solve cubic equations, Cartan encouraged his student Lodovico Ferrari (1522 – 1565) to use those result and solve quartic equations x4 +p x2 +q x +r=0. Since Tartaglia didn’t publish his results and after hearing from Hannibal Della Nave that Scipione del Ferro first solve cubic equations, Cardan pubished in 1545 in Ars Magna (The Great Art) the solutions of the cubic (credit given to Tartaglia) and quartic equations. This led to another competition between Tartaglia and Cardano, for which the latter did not show up but was represented by his student Lodovico Ferrari. Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and income.
19 SOLO Complex Variables History of Complex Numbers Solution of cubic equation x3 + b x2 +c x +d = 0 023 =+++ dxcxbx 3/bxy −= → 0 27 2 33 393 2 273333 33 2 3 3 22 32 23 23 =−+=      +−+      −+= +−++−+−+−=+      −+      −+      − − nymyb cb dy b cy d cb yc b ybyb b y b ybyd b yc b yb b y nm    Equivalence between x3 + b x2 +c x +d = 0 and y3 +m y = n (depressed cubic equation) Solutions of y3 +m y = n Start from the identity: ( )  ( ) nymybabababa nymy =+→−=−+− 3333 3  0 27273 3 3 36 3 3 333 =−−→−=−==→= m ana a m aban a m bbam 3 32 3,2,13,2,1 2742 mnn wa +±=                       +± −      +±=−= 3 32 3 32 3,2,13,2,1 342 3 3423 mnn m mnn w a m ay 2 31 , 2 31 ,13 3,2,1 ii ew ki k +− === π
20 SOLO Complex Variables History of Complex Numbers Solution of cubic equation x3 + b x2 +c x +d = 0                 ++      +±=                       −      −            + −      +±=                       +± −      +±=−= 3 32 3 32 3,2,1 3 322 3 32 3 32 3,2,1 3 32 3 32 3,2,13,2,1 342342 322 342 3342 342 3 3423 mnnmnn w mnn mnn mmnn w mnn m mnn w a m ay   2 31 , 2 31 ,1 342342 3 3,2,1 3 32 3 32 3,2,13,2,1 ii ew mnnmnn wy ki k +− ==                 +−+      ++= = π Solutions of y3 +m y = n Note: Tartaglia and Cardano knew only the solution w1 = 1
21 SOLO Complex Variables History of Complex Numbers Solution of cubic equation x3 + b x2 +c x +d = 0 A Solution of y3 +m y = n Guess a solution:       −++= 33 huhuwy ( ) uzhuhuhuhuhuhuy 233 3 2333 23 +−=−+      −++−++= Therefore:              +      = = →     −=− = 32 3 2 32 2 3 2 mn h n u mhu nu                 +      −+      +      += 3 32 3 32 3,2,13,2,1 322322 mnnmnn wy Where w are the roots of w3 =1: 2 31 , 2 31 ,13,2,1 ii w +− =
22 SOLO Complex Variables History of Complex Numbers Solution of cubic equation x3 + b x2 +c x +d = 0 Viète Solution of y3 +m y = n François Viète 1540 - 1603 In 1591 François Viète gave another solution to y3 +m y = n Start with identity CC C 34cos3cos43cos 3 cos 3 −=−= = θ θθθ Substitute in y3 +m y = nC m y 3 2 −= 3 3 3 2 1 3 3 3 2 34 3 2 3 8 3       −=−→=−+      −       − m n CCnC m mC m m Assuming we obtain 323 32 1 3 2         ≤      →≤      mn m n φθ πφθ cos 3 2 3cos 23 3 k m n += =      −=               −=      + −=−= − 3 1 3 2 cos 3 2 cos 3 2 3 2 m nkm C m y φ πφ
23 SOLO Complex Variables History of Complex Numbers Solution of cubic equation x3 + b x2 +c x +d = 0 Comparison of Cardano and Viète Solution of y3 +m y = n François Viète 1540 - 1603Cardano solution was               −=      + −=+= − 3 1 3 2 cos 3 2 cos 3 2 m nkm ssy φ πφ Girolamo Cardano 1501 - 15763 32 3 32 342342       +−+      ++= mnnmnn y ( )                              −=       −=       ++= += − + 3 1 2 3 32 3 3 2 cos 3 342 33 m n e m mnn s ssy ki ss φ πφ  or from which we recover Viète Solution 0s φ 2 n 3 3      m 3 s 3 s 0 x 3/φ 1s 2s 0 s 2s 1 s 23 23       −      nm 3 z  120  120  120
24 SOLO Complex Variables History of Complex Numbers Rafael Bombelli 1526 - 1572 John Wallis 1616 - 1703 In 1572 Rafael Bombelli published three of the intended five volumes of “L’Algebra” worked with non-real solutions of the quadratic equation x2 +b x+c=0 by using and where and applying addition and multiplication rules. vu 1−+ vu 1−− ( ) 11 2 −=− In 1673 John Wallis presented a geometric picture of the complex numbers resulting from the equation x2 + b x + c=0, that is close with what we sed today. Wallis's method has the undesirable consequence that is represented by the same point as is 1−− 1− ( )0,b−( )0,b− bb c bb c1 P 2P 2 P 1 P Wallis representation of real roots of quadratics Wallis representation of non-real roots of quadratics
25 Vector Analysis HistorySOLO Caspar Wessel 1745-1818 “On the Analytic Representation of Direction; an Attempt”, 1799 bia + Jean Robert Argand 1768-1822 1806 1−=i 3.R.S. Elliott, “Electromagnetics”,pp.564-568 http://www-groups.dcs.st-and.ac.uk/~history/index.html Wessel's fame as a mathematician rests solely on this paper, which was published in 1799, giving for the first time a geometrical interpretation of complex numbers. Today we call this geometric interpretation the Argand diagram but Wessel's work came first. It was rediscovered by Argandin 1806 and again by Gauss in 1831. (It is worth noting that Gauss redid another part of Wessel's work, for he retriangulated Oldenburg in around 1824.)
26 SOLO Complex Variables History of Complex Numbers Leonhard Euler 1707- 1783 In 1748 Euler published “Introductio in Analysin Infinitorum” in which he introduced the notation and gave the formula1−=i xixeix sincos += In 1751 Euler published his full theory of logarithms and complex numbers. Euler discovered the Cauchy-Riemann equations in 1777 although d’Alembert had discovered them in 1752 while investigating hydrodynamics. Johann Karl Friederich Gauss published the first correct proof of the fundamental theorem of algebra in his doctoral thesis of 1797, but still claimed that "the true metaphysics of the square root of -1 is elusive" as late as 1825. By 1831 Gauss overcame some of his uncertainty about complex numbers and published his work on the geometric representation of complex numbers as points in the plane. Karl Friederich Gauss 1777-1855
27 SOLO Complex Variables History of Complex Numbers Augustin Louis Cauchy )1789-1857( Cauchy is considered the founder of complex analysis after publishing the Cauchy-Riemann equations in 1814 in his paper “Sur les Intégrales Définies”. He created the Residue Theorem and used it to derive a whole host of most interesting series and integral formulas and was the first to define complex numbers as pairs of real numbers. Georg Friedrich Bernhard Riemann 1826 - 1866 In 1851 Riemann give a dissertation in the theory of functions. Return to Table of Contents
28 SOLO Complex Variables Derivatives If f (z) is a single-valued in some region C of the z plane, the derivative of f (z) is defined as: ( ) ( ) ( ) z zfzzf zf z ∆ −∆+ = →∆ 0 lim' provided that the limit exists independent of the manner in which Δ z→0. In such case we say that f (z) is differentiable at z. Analytic Functions If the derivative of f (z) exists at all points of a region C of the z plane, then f (z) is said to be analytic in C. analytic = regular = holomorphic A function f (z) is said to be analytic at a point z0 if there exists a neighborhood |z-z0 | < δ in which f ’ (z) exists. z0 δ Analytic functions have derivatives of any order which themselves are analytic functions. Return to Table of Contents
29 SOLO Complex Variables Derivatives If f (z) is a single-valued in some region C of the z plane, the derivative of f (z) is defined as: ( ) ( ) ( ) z zfzzf zf z ∆ −∆+ = →∆ 0 lim' provided that the limit exists independent of the manner in which Δ z→0. In such case we say that f (z) is differentiable at z. Analytic Functions If the derivative of f (z) exists at all points of a region C of the z plane, then f (z) is said to be analytic in C. analytic = regular = holomorphic A function f (z) is said to be analytic at a point z0 if there exists a neighborhood |z-z0 | < δ in which f ’ (z) exists. z0 δ Analytic functions have derivatives of any order which themselves are analytic functions.
30 SOLO Complex Variables Analytic, Holomorphic, MeromorphicFunctions Return to Table of Contents A Meromorphic Function on an open subset D of the complex plane is a function that is Holomorphic on all D except a set of isolated points, which are poles for the function. (The terminology comes from the Ancient Greek meros (μέρος), meaning “part”, as opposed to holos ( λος)ὅ , meaning “whole”.) The word “Holomorphic" was introduced by two of Cauchy's students, Briot (1817–1882) and Bouquet (1819–1895), and derives from the Greek λοςὅ (holos) meaning "entire", and μορφή (morphē) meaning "form" or "appearance".[2] Today, the term "holomorphic function" is sometimes preferred to "analytic function", as the latter is a more general concept. This is also because an important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow directly from the definitions. The term "analytic" is however also in wide use. The Gamma Function is Meromorphic in the whole complex plane Poles
31 SOLO Complex Variables Cauchy-Riemann Equations A necessary (but not sufficient) condition that f (z) = u (x,y) +i v (x,y) be analytic in a region C, is that u and v satisfy the Cauchy-Riemann equations: y u x v y v x u ∂ ∂ −= ∂ ∂ ∂ ∂ = ∂ ∂ & Proof: Augustin Louis Cauchy )1789-1857( Georg Friedrich Bernhard Riemann 1826 - 1866 ( ) ( ) ( ) z zfzzf zf z ∆ −∆+ = →∆ 0 lim' Provided that the limit exists independent of the manner in which Δ z→0. Choose Δ z = Δ x → ( ) x v i x u zf ∂ ∂ + ∂ ∂ =' Choose Δ z =i Δ y → ( ) y v y u izf ∂ ∂ + ∂ ∂ −=' Equalizing those two expressions we obtain: ( ) y u i y v x v i x u zf ∂ ∂ − ∂ ∂ = ∂ ∂ + ∂ ∂ =' y u x v y v x u ∂ ∂ −= ∂ ∂ ∂ ∂ = ∂ ∂ → & The functions u (x,y) and v (x,y) are called conjugate functions, because if one is given we can find the other (with an arbitrary additive constant). Return to Table of Contents
32 SOLO Complex Variables Harmonic Functions If the second partial derivatives of u (x,y) and v (x,y) with respect to x and y exist and are continuous in a region C, then using Cauchy-Riemann equations, we obtain: 02 2 2 2 2 22 2 2 2 22 = ∂ ∂ + ∂ ∂ →         ∂ ∂ −= ∂∂ ∂ → ∂ ∂ −= ∂ ∂ ∂∂ ∂ = ∂ ∂ → ∂ ∂ = ∂ ∂ ∂∂ ∂ = ∂∂ ∂ ∂ ∂ ∂ ∂ y u x u y u xy v y u x v yx v x u y v x u xyyx y x 02 2 2 2 2 2 2 2 22 22 = ∂ ∂ + ∂ ∂ →         ∂∂ ∂ −= ∂ ∂ → ∂ ∂ −= ∂ ∂ ∂ ∂ = ∂∂ ∂ → ∂ ∂ = ∂ ∂ ∂∂ ∂ = ∂∂ ∂ ∂ ∂ ∂ ∂ y v x v yx u x v y u x v y v xy u y v x u xyyx x y It follows that under those conditions the real and imaginary parts of an analytic function satisfy Laplace’s Equation denoted by: 02 2 2 2 = ∂ Φ∂ + ∂ Φ∂ yx or 2 2 2 2 2 : yxx i xx i x ∂ ∂ + ∂ ∂ =      ∂ ∂ − ∂ ∂ ⋅      ∂ ∂ + ∂ ∂ =∇ where02 =Φ∇ The functions satisfying Laplace’s Equation are called Harmonic Functions. Pierre-Simon Laplace (1749-1827) Return to Table of Contents
33 SOLO Complex Variables Singular Points A point at which f (z) is not analytic is called a singular point. There are various types of singular points: 1. Isolated Singularity The point z0 at which f (z) is not analytic is called an isolated singular point, if we can a neighborhood of z0 in which there are not singular points. z0 δ If no such a neighborhood of z0 can be found then we call z0 a non-isolated singular point. 2. Poles Example ( ) ( ) ( ) ( ) ( ) ( ) ( )5353 1834 32 ++−− ++ = zzzz zz zf has a pole of order 2 at z = 3, a pole of order 3 at z = 5, and two simple poles at z = -3 and z = -5. If we can find a positive integer n such that and is analytic at z=z0 then z = z0 is called a pole of order n. If n = 1, z is called a simple pole. ( ) ( ) 0lim 0 0 ≠=−→ Azfzz n zz ( ) ( ) ( )zfzzz n 0−=ϕ
34 SOLO Complex Variables Singular Points A point at which f (z) is not analytic is called a singular point. There are various types of singular points: 3. Branch Points If f (z) is a multiple valued function at z0, then this is a branch point. Examples: ( ) ( ) n zzzf /1 0−= has a branch point at z=z0 ( ) ( ) ( )[ ]0201ln zzzzzf −−= has a branch points at z=z01 and z=z02 4. Removable Singularities The singular point z0 is a removable singularity of f (z) if exists.( )zf zz 0 lim → Examples: The singular point z = 0 of is a removable singularity z zsin 1 sin lim0 =→ z z z
35 SOLO Complex Variables Singular Points A point at which f (z) is not analytic is called a singular point. There are various types of singular points: 5. Essential Singularities A singularity which is not a pole, branch point or a removable singularity is called an essential singularity. Example: has an essential singularity at z = z0.( ) ( )0/1 zz ezf − = 6. Singularities at Infinity If we say that f (z) has singularities at z →∞. The type of the singularity is the same as that of f (1/w) at w = 0. ( ) 0lim = ∞→ zf z Example: The function f (z) = z5 has a pole of order 5 at z = ∞, since f (1/w) = 1/w5 has a pole of order 5 at w = 0. Return to Table of Contents
36 SOLO Complex Variables Orthogonal Families If f (z) = u (x,y) + i v (x,y) is analytic, then the one-parameter families of curves ( ) ( ) βα == yxvyxu ,,, where α and β are constant are orthogonal. Proof: The normal to u (x,y) = α is: ( ) y y u x x u yxu 11, ∂ ∂ + ∂ ∂ =∇ The normal to v (x,y) = β is: ( ) y y v x x v yxv 11, ∂ ∂ + ∂ ∂ =∇ The scalar product between the normal to u (x,y) = α and the normal to v (x,y) = β is: ( ) ( ) y v y u x v x u yxvyxu ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ =∇⋅∇ ,, Using the Cauchy-Riemann Equation for the analytic f (z): y u x v y v x u ∂ ∂ −= ∂ ∂ ∂ ∂ = ∂ ∂ & ( ) ( ) 0,, = ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ −=∇⋅∇ y v y u y u y v yxvyxu x y ( ) α=yxu , ( ) β=yxv , planez u v planew Return to Table of Contents
37 SOLO Complex Variables Complex Line Integrals Let f (z) be continuous at all points on a curve C of a finite length L. ( ) ( ) ( )∑∑ == − ∆=−= n i ii n i iiin zfzzfS 11 1 ξξ C 1 z nzb = 2z 0 za = 1−iz iz 1 ξ 2 ξ i ξ n ξ Let subdivide C into n parts by n arbitrary points z1, z2,…,zn, and call a=z0 and b=zn. On each arc joining zi-1 to zi choose a point ξi. Define the sum: Let the number of subdivisions n increase in such a way that the largest of Δzi approaches zero, then the sum approaches a limit that is called the line integral (also Riemann-Stieltjes integral). ( ) ( ) ( )∫∫∑ ==∆= = →∆∞→ C b a n i ii z nn zdzfzdzfzfS i 1 0 limlim ξ Properties of Integrals ( ) ( )[ ] ( ) ( )∫∫∫ +=+ CCC zdzgzdzfzdzgzf ( ) ( ) constantAzdzfAzdzfA CC == ∫∫ ( ) ( )∫∫ −= a b b a zdzfzdzf ( ) ( ) ( )∫∫∫ += b c c a b a zdzfzdzfzdzf ( ) ( ) ( ) CoflengthLandConMzfLMzdzfzdzf CC ≤≤≤ ∫∫ Return to Table of Contents
38 SOLO Complex Variables Simply and Multiply Connected Regions A region R is called simply-connected if any simple closed curve Γ, which lies in R can be shrunk to a point without leaving R. A region R that is not simply-connected is called multiply-connected. C0 x y R C1 Γ C0 x y R C1 C2 C3 Γ C x y R Γ C x y R Γsimply-connected multiply-connected. Return to Table of Contents
39 SOLO Complex Variables Green’s Theorem in the Plane C R Let P (x,y) and Q (x,y) be continuous and have continuous partial derivatives in a region R and on the boundary C. Green’s Theorem states that: GEORGE STOCKES 1819-1903 A more general theorem was given by Stokes ( ) ∫∫∫       ∂ ∂ − ∂ ∂ =+ R dydx y P x Q dyQdxP C ( ) ∫∫∫∫∫∫∫       ∂ ∂ − ∂ ∂ +      ∂ ∂ − ∂ ∂ +      ∂ ∂ − ∂ ∂ =++ yzxzxy RRR dzdy z Q y R dzdx x R z P dydx y P x Q dzRdyQdxP C or in vector form: ∫∫∫ ⋅×∇=⋅ S dAFdrF C  where: ( ) ( ) ( ) ( ) zzyxRyzyxQxzyxPzyxF 1,,1,,1,,,, ++=  zdzydyxdxdr 111 ++= zdydxydzdxxdzdydA 111 ++= GEORGE GREEN 1793-1841 z z y y x x 111 ∂ ∂ + ∂ ∂ + ∂ ∂ =∇
40 SOLO Complex Variables Proof of Green’s Theorem in the Plane C R P T S Q a b x y ( )xgy 2= ( )xgy 1= Start with a region R and the boundary curve C, defined by S,Q,P,T, where QP and TS are parallel with y axis. ( ) ( ) ∫ ∫∫∫ = = ∂ ∂ = ∂ ∂ b a xgy Xgy dy y P dxdydx y P 2 R By the fundamental lemma of integral calculus: ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )[ ]xgxPxgxPyxPdy y yxP xgy xgy xgy Xgy 12 ,,, , 2 1 2 −== ∂ ∂ = = = = ∫ Therefore: ( )[ ] ( )[ ]∫∫∫∫ −= ∂ ∂ b a b a dxxgxPdxxgxPdydx y P 12 ,, R but: ( )[ ] ( )[ ]∫∫ = a bSQ dxxgxPdxxgxP 22 ,, integral along curve SQ ( )[ ] ( )[ ]∫∫ = b aPT dxxgxPdxxgxP 11 ,, integral along curve PT If we add to those integrals: ( ) ( ) 00,, === ∫∫ dxsincedxyxPdxyxP QPTS we obtain: ( )[ ] ( ) ( )[ ] ( ) ( )∫∫∫∫∫∫∫ −=−−−−= ∂ ∂ CTSPTQPSQ dxyxPdxyxPdxxgxPdxyxPdxxgxPdydx y P ,,,,, 12 R Assume that PT is defined by the function y = g1 (x) and SQ is defined by the function y = g2 (x), both smooth and y P ∂ ∂ is continuous in R:
41 SOLO Complex Variables Proof of Green’s Theorem in the Plane (continue – 1) In the same way: Therefore we obtain: ( )∫∫∫ −= ∂ ∂ C dxyxPdydx y P , R ( )∫∫∫ = ∂ ∂ C dyyxQdydx x Q , R ( ) ∫∫∫       ∂ ∂ − ∂ ∂ =+ R dydx y P x Q dyQdxP C The line integral is evaluated by traveling C counterclockwise. For a general single connected region, as that described in Figure to the right, can be divided in a finite number of sub-regions Ri, each of each are of the type described in the Figure above. Since the adjacent regions boundaries are traveled in opposite directions, there sum is zero, and we obtain again: ( ) ∫∫∫       ∂ ∂ − ∂ ∂ =+ R dydx y P x Q dyQdxP C C R4 x y R R3 R1 R2 C R P T S Q a b x y ( )xgy 2= ( )xgy 1=
42 SOLO Complex Variables Proof of Green’s Theorem in the Plane (continue – 2) The general multiply-connected regions can be transformed in a simply connected region by infinitesimal slits Since the slits boundaries are traveled in opposite directions, there integral sum is zero: C0 x y R C1 P0 P1 C0 x y R C1 C2 C3 ( ) ( ) ∫∫∑ ∫∫       ∂ ∂ − ∂ ∂ =+−+ R dydx y P x Q dyQdxPdyQdxP i CC i0 All line integrals are evaluated by traveling Ci i=0,1,… counterclockwise. ( ) ( ) 0 0 1 1 0 =+++ ∫∫ P P P P dyQdxPdyQdxP We obtain: Return to Table of Contents
43 SOLO Complex Variables Consequences of Green’s Theorem in the Plane Let P (x,y) and Q (x,y) be continuous and have continuous first partial derivative at each point of a simply-connected region R. A necessary and sufficient condition that around every closed path C in R is that in R. This is synonym to the condition that is path independent. y P x Q ∂ ∂ = ∂ ∂ ( ) 0=+∫C dyQdxP Sufficiency: Suppose y P x Q ∂ ∂ = ∂ ∂ According to Green’s Theorem ( ) 0=      ∂ ∂ − ∂ ∂ =+ ∫∫∫ R dydx y P x Q dyQdxP C Necessity: 0<> ∂ ∂ − ∂ ∂ or y P x Q Suppose along every path C in R. Assume that at some point (x0,y0) in R. Since ∂ Q/ ∂ x and ∂ P/ ∂ y are continuous exists a region τ around (x0,y0) and boundary Γ for which , therefore: ( ) 0=+∫C dyQdxP 0<> ∂ ∂ − ∂ ∂ or y P x Q ( ) 0<>      ∂ ∂ − ∂ ∂ =+ ∫∫∫Γ ordydx y P x Q dyQdxP τ C x y R ( )∫ + L dyQdxP 0= ∂ ∂ − ∂ ∂ y P x Q This is a contradiction to the assumption, therefore q.e.d. Return to Table of Contents
44 SOLO Complex Variables Cauchy’s Theorem C x y R Proof: ( ) 0=∫C dzzf If f (z) is analytic with derivative f ‘ (z) which is continuous at all points inside and on a simple closed curve C, then: ( ) ( ) ( )yxviyxuzf ,, +=Since is analytic and has continuous first order derivative ( ) y u i y v x v i x u zd fd zf iyzxz ∂ ∂ − ∂ ∂ = ∂ ∂ + ∂ ∂ == == ' y u x v y v x u ∂ ∂ −= ∂ ∂ ∂ ∂ = ∂ ∂ & Cauchy - Riemann ( ) ( ) ( ) ( ) ( ) 0 00 =      ∂ ∂ − ∂ ∂ +      ∂ ∂ − ∂ ∂ −= ++−=++= ∫∫∫∫ ∫∫∫∫ RR dydx y v x u idydx y u x v dyudxvidyvdxudyidxviudzzf CCCC    q.e.d. Augustin Louis Cauchy )1789-1857( Return to Table of Contents
45 SOLO Complex Variables Cauchy-Goursat Theorem C x y R Proof: ( ) 0=∫C dzzf If f (z) is analytic which is continuous at all points inside and on a simple closed curve C, then: Augustin Louis Cauchy )1789-1857( Goursat removed the Cauchy’s condition that f ‘ (z) should be continuous in R. C F DE A B I ∆ IV ∆ II ∆ III ∆ Start with a triangle ABC in z in which f (z) is analytic, Join the midpoints E,D,F to obtain four equal triangles ΔI, ΔII, ΔIII, ΔIV. We have: ( ) ( ) ( ) ( ) ( )∫∫∫∫ ∫∫∫∫ ∫∫∫∫∫∫∫∫∫ ∫∫∫∫ ∆∆∆∆ +++= +++=         +++         ++         ++         += ++= IVIIIIII dzzfdzzfdzzfdzzf dzzf DEFDFCDFEBFEDAED FDEFDEDFFCDFEEBFEDDAE FCDEBFDAEABCA Eduard Jean-Baptiste Goursart 1858 - 1936
46 SOLO Complex Variables Proof of Cauchy-Goursat Theorem (continue – 1) If f (z) is analytic which is continuous at all points inside and on a simple closed curve C, then: C F DE A B I ∆ IV∆ II ∆ III ∆ ( ) ( ) ( ) ( ) ( )∫∫∫∫∫ ∆∆∆∆ +++= IVIIIIII dzzfdzzfdzzfdzzfdzzf ABCA then: ( ) ( ) ( ) ( ) ( )∫∫∫∫∫ ∆∆∆∆ +++≤ IVIIIIII dzzfdzzfdzzfdzzfdzzf ABCA Let Δ1 be the triangle in which the absolute value of the integral is maximum. ( ) ( )∫∫ ∆∆ ≤ 1 4 dzzfdzzf Continue this procedure in triangle Δ1 in which Δ2 is the triangle in which the absolute value of the integral is maximum. ( ) ( ) ( )∫∫∫ ∆∆∆ ≤≤ 21 2 44 dzzfdzzfdzzf ( ) ( )∫∫ ∆∆ ≤ n dzzfdzzf n 4
47 SOLO Complex Variables Proof of Cauchy-Goursat Theorem (continue – 2) C F DE A B I ∆ IV ∆ II ∆ III ∆ ( ) ( )∫∫ ∆∆ ≤ n dzzfdzzf n 4 For an analytic function f (z) compute ( ) ( ) ( ) ( )0 0 0 0 ':, zf zz zfzf zz − − − =η ( ) ( ) ( ) ( ) ( ) ( ) 0'''lim,lim 000 0 0 0 00 =−=       − − − = →→ zfzfzf zz zfzf zz zzzz η ( ) ( ) ( ) ( ) ( ) ( ) ( )0000000 ,'&,..,0 zzzzzzzfzfzfzzwheneverzzts −+−+=<−<∃>∀ ηδεηδε ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )∫∫∫∫ ∆∆ ← ∆∆ −≤−+−+≤ nnnn dzzzzzdzzzzzdzzzzfzfdzzf TheoremIntegralCauchy 0000 0 )00 ,,' ηη    n∆ 0z na nb nc z 0 zz − 0 zzcbaP nnnn −≥++= ( ) ( ) ( ) 2 2 00 2 ,       ==≤−≤ ∫∫∫ ∆∆∆ nnn P PdzPdzzzzzdzzf nnn εεεη But , where Pn the perimeter of Δn and P the perimeter of Δ are related, by construction, by ( ) δεη <≤−< n Pzzzz 00 &, n n PP 2/= q.e.d. ( ) ( ) ( ) 0 4 44 0 2 2 =→=≤≤ ∫∫∫ ∆ → ∆∆ dzzfP P dzzfdzzf n nn n ε εε
48 SOLO Complex Variables Proof of Cauchy-Goursat Theorem (continue – 3) n z1 z 2 z 1−i z i z 1−n z n ∆1∆ 2∆ 3 ∆ i ∆ C O q.e.d. For the general case of a simple closed curve C we take n points on C: z1, z2,…,zn and a point O inside C. We obtain n triangles Δ1, Δ2,.., Δn, for each of them we proved Cauchy-Goursat Theorem. Let define the sum: ( ) ∑= − − ∆= n i zz iin ii zzfS 1 1 : we have: ( ) ( ) ( ) ( ) 0 1 1 =++= ∫∫∫∫ − −∆ i i i ii z O O z z z dzzfdzzfdzzfdzzf ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( )    n i i i i i i i ii S n i z z i n i z z i n i z z ii n i z z n i dzzfdzzfzfdzzfzfzfdzzfdzzf ∑ ∫∑∫∑ ∫∑ ∫∑ ∫ ===== ∆ −−−− +−=+−=== 11111 1111 0 ( ) ( ) ( )ε ε NnforSdzzfdzzfS n CC nn ><−→= ∫∫∞→ 2 lim ( ) ( )[ ] ( ) ( )[ ] ( ) 221 1 111 11 εε =−≤−≤−≤→−= ∑∑∑ ∫∑∫ = − === −− n i ii n i i n i z z in n i z z in zz L dzfzfdzzfzfSdzzfzfS i i i i ( ) ( ) ( ) ( ) 0 22 =→>=+<+−≤ ∫∫∫ C nn CC dzzfNnforSSdzzfdzzf εε εε Since we proved that , we can write:( ) 0=∫∆ dzzf Return to Table of Contents
49 SOLO Complex Variables Consequences of Cauchy-Goursat Theorem B x y R A C1 C2 D1 D2 a bIf f (z) is analytic in a simply-connected region R, then is independent of the path in R joining any two points a and b in R. ( )∫ b a dzzf Let look at thr closed path AC1D1BD2C2A in R inside which f (z) is analytic. According to Cauchy-Goursat Theorem ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫∫∫∫∫ =→−=+== BDAcBDAcBDAcBDAcACBDBDAcACBDDAC dzzfdzzfdzzfdzzfdzzfdzzfdzzf 2211221122111122 0 Proof: If f (z) is analytic in a multiply-connected region R, bounded by two simple closed curves C1 and C2, then: 1 2 C1 x y R C2 P0 P1 ( ) ( )∫∫ = 21 CC dzzfdzzf The general multiply connected regions can be transformed in a single connected region by an infinitesimal slit P0 to P1. ( ) ( ) ( ) ( ) ( ) ( )∫∫∫∫∫∫ =→+++= 212 0 1 1 01 0 0 CCC P P P PC dzzfdzzfdzzfdzzfdzzfdzzf    Proof: Return to Table of Contents
50 SOLO Complex Variables Cauchy’s Integral Formulas Augustin Louis Cauchy )1789-1857( If f (z) is analytic inside and on a simple closed curve C and a is any point inside C then ( ) ( ) ∫ − = C dz az zf i af π2 1 C x y R a Γ Proof: Let chose a circle Γ with center at a [ ]{ }πθε θ 2,0,: ∈+==Γ i eazz Since f (z)/ (z-a) is analytic in the region defined between C and the circle Γ we can use: ( ) ( ) ∫∫ Γ − = − zd az zf zd az zf C ( ) ( ) ( ) ( )afidafidei e eaf zd az zf i i i πθθε ε ε ππ θ θ θ ε 21lim 2 0 2 0 0 === + = − ∫∫∫ → Γ therefore: ( ) ( ) ∫ − = C dz az zf i af π2 1 q.e.d. Cauchy’s Integral Formulas and Related Theorems Return to Table of Contents
51 SOLO Complex Variables Cauchy’s Integral Formulas for the n Derivative of a Function Augustin Louis Cauchy )1789-1857( If f (z) is analytic inside and on a simple closed curve C and a is any point inside C, where the n derivative exists, then ( ) ( ) ( ) ( )∫ + − = C n n dz az zf i n af 1 2 ! π C x y R a Γ Proof: Let prove this by induction. Assume that this is true for n-1: Then we can differentiate under the sign of integration: ( ) ( ) ∫ − = C dz az zf i af π2 1 For n = 0 we found ( ) ( ) ( ) ( ) ( )∫ − − =− C n n dz az zf i n af π2 !11 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ + − − − = − − == C n C n nn dz az n zf i n dz azad d zf i n af ad d af 1 1 2 !11 2 !1 ππ q.e.d.Therefore for n we obtain: ( ) ( ) ( ) ( )∫ + − = C n n dz az zf i n af 1 2 ! π We can see that an analytic function has derivatives of all orders. Return to Table of Contents
52 SOLO Complex Variables Morera’s Theorem (the converse of Cauchy’s theorem) If f (z) is continuous in a simply-connected region R and if around every simple closed curve C in R then f (z) ia analytic in R. ( ) 0=∫C dzzf B x y R A C1 C2 D1 D2 a z Proof: Since around every closed curve C in R( ) 0=∫C dzzf ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫∫∫∫∫ =→−=+== BDAcBDAcBDAcBDAcACBDBDAcACBDDAC dzzfdzzfdzzfdzzfdzzfdzzfdzzf 2211221122111122 0 The integral is independent on path between two points, if the path is in R ( ) ( )∫= z a dzzfzF Let choose a straight path between z and z+Δz ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∫∫∫ ∆+∆+ − ∆ =−         − ∆ =− ∆ −∆+ zz z z a zz a udzfuf z zfudufuduf z zf z zFzzF 11 Since f (z) is continuous ( ) ( ) δε <−≤− zuwheneverzfuf Therefore ( ) ( ) ( ) ( ) ( ) ( ) ( )zf zd zFd zudzfuf z zf z zFzzF zz z =→<∆∀=− ∆ ≤− ∆ −∆+ ∫ ∆+ δε 1 C x y R z Νδz+∆ z Since F (z) has a derivative in R, it is analytic, and so are its derivatives, i.e. f (z) Return to Table of Contents Giacinto Morera 1856 - 1907
53 SOLO Complex Variables Cauchy’s Inequality Augustin Louis Cauchy )1789-1857( If f (z) is analytic inside and on a circle C of radius r and center at z = a, then ( ) ( ) ,...2,1,0 ! = ⋅ ≤ n r nM af n n where M is a constant such that | f (z) |< M is an upper bound of | f (z) | on C. Proof: C x y a r Use Cauchy Integral Formula: ( ) ( ) ( ) ( ) ,2,1,0 ! 2 1 2 !1 2 ! 2 ! 1 2 0 11 ===≤ − ≤ ++ < + ∫∫ n r nM r r Mn dr r Mn dz az zfn af nnn Mzf C n n π π θ ππ π ( ) ( ) ( ) ( ) ,2,1,0 2 ! 1 = − = ∫ + ndz az zf i n af C n n π q.e.d. On the circle C: .θi eraz += Return to Table of Contents
54 SOLO Complex Variables Liouville’s Theorem Joseph Liouville 1809 - 1882 If for all z in the complex plane: (1) f (z) is analytic (2) f (z) is bounded, i.e. | f (z) |< M for some constant M then f (z) must be a constant. Proof No. 1: Using Cauchy’s Inequality: ( ) ( ) ,...2,1,0 ! = ⋅ ≤ n r nM af n n Letting n=1 we obtain: ( ) r M af ≤' Since f (z) is analytic in all z plane we can take r → ∞ to obtain ( ) ( ) ( ) constantafaaf r M af r =→∀=→=≤ ∞→ ,00lim '' q.e.d.
55 SOLO Complex Variables Liouville’s Theorem Joseph Liouville 1809 - 1882 If for all z in the complex plane: (1) f (z) is analytic (2) f (z) is bounded, i.e. | f (z) |< M for some constant M then f (z) must be a constant. Proof No. 2: Using Cauchy’s Integral Formula q.e.d. Let a and b be any two points in z plane. Draw a circle C with center at a and radius r > 2 | a-b | C x y a r b 2/rba <− ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ −− − = − − − =− CCC dz bzaz zf i ab dz az zf i dz bz zf i afbf πππ 22 1 2 1 We have raz =− ( ) ( ) 2/rbarbaazbaazbz ≤−−=−−−≥−+−=− ( ) ( ) ( ) ( ) ( ) ( ) ( ) r Mab dr rr Mab dz bzaz zfab dz bzaz zfab afbf CC − = − ≤ −− − ≤ −− − =− ∫∫∫ 2 2/222 2 0 π θ πππ Since f (z) is analytic in all z plane we can take r → ∞ to obtain ( ) ( ) ( ) ( )afbfafbf =→=− 0 therefore f (z) is constant. Return to Table of Contents
56 SOLO Complex Variables Foundamental Theorem of Algebra Every polynomial equation P (z) = a0 + a1z+a2z2 +…+anzn =0 with degree n ≥ 1 and an ≠ 0 (ai are complex constants) has at least one root. From this it follows that P (z) = 0 has exactly n roots, due attention being paid to multiplicities of roots. Proof: If P (z) = 0 has no root, then f (z) = 1 / P (z) is analytic for all z. Also | f (z) |= 1 / | P (z) | is bounded. Then by Liouville’s Theorem f (z) and then P (z) are constant. This is a contradiction to the fact that P (z) is a polynomial in z, therefore P (z) = 0 must have at least one root (zero). Suppose that z = a is one root of P (z) = 0. Hence P (a) = 0 and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )zQazazaazaaza aaaaaaazazazaaaPzP nn n n n n n −=−++−+−= ++++−++++=−   22 21 2 210 2 210 Since an ≠ 0, Q (s) is a polynomial of degree n-1. Applying the same reasoning to the polynomial Q (s) of degree n-1, we conclude that it must also have at least one root. This procedure continues until n = 0, therefore it follows that P (z) has exactly n roots. q.e.d. Return to Table of Contents
57 SOLO Complex Variables Gauss’ Mean Value Theorem Karl Friederich Gauss 1777-1855 C x y a r If f (z) is analytic inside and on a circle C with center at a and radius r, then f (a) is the mean of the values of f (z) on C, i.e., ( ) ( ) . 2 1 2 9 ∫ += π θ θ π derafaf i Proof: Use Cauchy Integral Formula: On the circle C: θθ ii eridzeraz =+= . ( ) ( ) ∫ − = C dz az zf i af π2 1 ( ) ( ) ( ) ( )∫∫∫ += + = − = θ π θ ππ θθ θ θ derafderi er eraf i dz az zf i af ii i i C 2 1 2 1 2 1 q.e.d. Return to Table of Contents
58 SOLO Complex Variables Maximum Modulus Theorem If f (z) is analytic inside and on a simple closed curve C and is not identically equal to a constant, then the maximum value of | f (z) | occurs on C. Proof: The proof is based on the continuity of f (z) and on the Gauss’ Mean Value Theorem. C x y a r C C1 C2 C3 R α b Since f (z) is analytic in C, | f (z) | has a maximum M inside or on C. Suppose that the maximum value is achieved at the point a inside C, i.e. | f (a) |=M =max | f (z) |. Since a is inside C we can find a circle C1, with center at a that is inside C. Since f (z) is not constant we can find a point b in C1 such that | f (b)|=M – ε< | f (a)|. Using the continuity of f (z) we can find a circle around b, C2, ( ) ( ){ }δε <−<−= bzforbfzfzC 2/:2 ( ) ( ) 2/2/2/ εεεε −=+−=+< MMbfzf Now apply the Gauss’ Mean Value Theorem for point a and the circle with center at a passing trough b, C3. Define by α the arc of C3 inside C2.( ) ( ) . 2 1 2 9 ∫ += π θ θ π derafaf i ( ) ( ) ( ) ( ) ( ) ( ) π εα απ ππ α εθθθ π π α θ α ε θ π θ 4 2 22 2/. 2 1 2 9 2/ 2 9 −=−+−≤+++≤+== ∫∫∫ ≤−≤ M M MderafderafderafMaf M i M ii 
59 SOLO Complex Variables Maximum Modulus Theorem If f (z) is analytic inside and on a simple closed curve C and is not identically equal to a constant, then the maximum value of | f (z) | occurs on C. Proof (continue): ( ) ( ) ( ) ( ) ( ) ( ) π εα απ ππ α εθθθ π π α θ α ε θ π θ 4 2 22 2/. 2 1 2 9 2/ 2 9 −=−+−≤+++≤+== ∫∫∫ ≤−≤ M M MderafderafderafMaf M i M ii  We obtained that is impossible, therefore a for which |f (z)| is maximum cannot be inside C, but on C. ( ) π εα 4 ? −== MMaf C x y a r C C1 C2 C3 R α b q.e.d. Return to Table of Contents
60 SOLO Complex Variables Minimum Modulus Theorem If f (z) is analytic inside and on a simple closed curve C and f (z) ≠ 0 inside C then | f (z) | assumes its minimum value on C. Proof: Since f (z) is analytic inside and on a simple closed curve C and f (z) ≠ 0 inside C it follows that 1/ f (z) is analytic inside and on C. Then according to Maximum Modulus Theorem 1/| f (z) | assumes its maximum vale on C and therefore | f (z) | assumes its minimum value on C. q.e.d. x y C R Return to Table of Contents
61 SOLO Complex Variables Poisson’s Integral Formulas for a Circle Siméon Denis Poisson 1781-1840 Let f (z) be analytic inside and on the circle C defined by |z| = R, and let z = r e iθ be any point inside C, then: ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ − − = − − = +−− − == π π φ φ φ θφ π φθ φ π φ π φ φθπ 2 0 2 0 2 22 2 22 2 0 22 22 2 1 2 1 cos22 1 deRf zeR zR deRf ereR rR deRf rrRR rR erzf i i i ii ii C x y R R z' rR2/r θ ∗ z z r θ− ∗ = zRz /2 1 Proof: Since f (z) is analytic in C we can apply Cauchy’s Integral Formula: ( ) ( ) ( ) ∫ − == C i dz zz zf i erfzf ' ' ' 2 1 π θ ( ) ∫ ∗ − = C dz zRz zf i ' /' ' 2 1 0 2 π If we subtract those equations we obtain: The inverse of the point z with respect to C is and lies outside C, therefore by Cauchy’s Theorem: ∗ = zRz /2 1 ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∗∗ −− − =      − − − == CC i dzzf zRzzz zRz i dzzf zRzzzi erfzf '' /'' / 2 1 '' /' 1 ' 1 2 1 2 2 2 ππ θ
62 SOLO Complex Variables Poisson’s Integral Formulas for a Circle Siméon Denis Poisson 1781-1840 Proof (continue): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ ∫ +−− − = −− − = −− − = −− − = −− − == −− + ∗ π φ π φ θφθφ π φ θφθφ φθ π φφ θφθφ θθ θ φ φθπ φ π φ π φ π π 2 0 22 22 2 0 22 2 0 22 2 0 2 2 2 2 cos22 1 2 1 2 1 / / 2 1 '' /'' / 2 1 deRf RrRr rR deRf ereRereR rR deRf eRerereR eRr deRieRf erReRereR erRer i dzzf zRzzz zRz i erfzf i i iiii i iiii i ii iiii ii C i Writing we have:( ) ( ) ( )θθθ ,, rviruerf i += ( ) ( ) ( ) ( )∫ +−− − = π φφ φθπ θ 2 0 22 22 , cos22 1 , dRu RrRr rR ru ( ) ( ) ( ) ( )∫ +−− − = π φφ φθπ θ 2 0 22 22 , cos22 1 , dRv RrRr rR rv q.e.d. C x y R R z' rR2/r θ ∗ z z r θ− ∗ = zRz /2 1 Return to Table of Contents
63 SOLO Complex Variables Poisson’s Integral Formulas for a Half Plane Siméon Denis Poisson 1781-1840 C x y R ∗ z z R Let f (z) be analytic in the upper half y ≥ 0 of the z plane and let z = (x + i y) any point in this upper half plane, then: ( ) ( ) ( )∫ +∞ ∞− +− = dw yxw wfy zf 22 Proof: Let C be the boundary of a a semicircle of radius R containing as an interior point, but does not contain yixz += yixz −=∗ Using Cauchy’s Integral Formula we have: ( ) ( ) ∫ − = C dw zw wf i zf π2 1 By subtraction we obtain: ( ) ∫ ∗ − = C dw zw wf iπ2 1 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ +− = +−−− −−+ = −− − =      − − − = ∗∗ CC CC dwwf yxw yi i dwwf yixwyixw yixyix i dwwf zwzw zz i dwwf zwzwi zf 22 2 2 1 2 1 2 111 2 1 ππ ππ
64 SOLO Complex Variables Poisson’s Integral Formulas for a Half Plane Siméon Denis Poisson 1781-1840 Proof (continue): Where Γ is the upper a semicircle of radius R. ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫ Γ + − +− + +− = +− = dwwf yxw y dwwf yxw y dwwf yxw yi i zf R R C 2222 22 11 2 2 1 ππ π If we take R→∞ we obtain: ( ) ( ) 0lim 1 22 = +−∫Γ ∞→ dwwf yxw y R π ( ) ( ) ( )∫ +∞ ∞− +− = dw yxw wfy zf 22 Therefore: ( ) ( ) ( )∫ +∞ ∞− +− = dw yxw wufy yxu 22 0, , ( ) ( ) ( )∫ +∞ ∞− +− = dw yxw wvfy yxv 22 0, , Writing and since w varies on x axis , and we have: ( ) ( ) ( )yxviyxuzf ,, += ( ) ( ) ( )0,0, wviwuwf += q.e.d. C x y R ∗ z z R Return to Table of Contents
65 SOLO Infinite Series Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1Convergence Definition: The series Sn converges to S as n →∞ if for all ε > 0 there exists an positive integer N such that If no such N exists then we say that the series diverges. NnallforSuSS n i in ><−=− ∑= ε 1 Convergence Theorem: The series Sn converges as n →∞ if and only if there exists an positive integer M such that If no such M exists then we say that the series diverges. 1 1 ><= ∑= NallforMuS N i iN If S is unknown we can use the Cauchy Criterion for convergence: for all ε > 0 there exists an positive integer N such that NmnallforuuSS m j j n i imn ><−=− ∑∑ == , 11 ε Augustin Louis Cauchy )1789-1857( A necessary (but not sufficient) condition for convergence is that lim i→∞ ui = 0 Return to the Table of Content Cauchy Convergence Criterion
66 SOLO Infinite Series Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Convergence Tests In term by term a series of terms 0 ≤ un ≤ an, in which the an form a convergent series, then is also convergent.∑n nu Return to the Table of Content
67 SOLO The Geometric Series ( ) ( ) ( ) ( ) ( ) ( ) r r a r rrarararaa r rS nn nG − − = − −⋅+++++ = − − − − 1 1 1 1 1 1 132 1  Multiply and divide by (1 – r) ( ) ∑ − = − − =+++++= 1 0 132 1 n i in nG rararararaaS  We can see that ( )      ≥∞ < −= − − = ∞→ − ∞→ diverger converger r a r r aS n n nG n 1 1 1 1 1 limlim 1 Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series
68 SOLO Convergence Tests Cauchy Root Test Augustin Louis Cauchy )1789-1857( If (an) 1/n ≤ r < 1 for all sufficiently large n, with r independent of n, then is convergent. If (an) 1/n ≥ 1 for all sufficiently large n, then is divergent. ∑n na ∑n na The first part of this test is verified easily by raising (an) 1/n ≤ r to the nth power. We get: 1<≤ n n ra ∑n naSince rn is just the nth term in a Convergent Geometric Series, is convergent by the Comparison Test. Conversely, if (an) 1/n ≥ 1, the an ≥ 1 and the series diverge. This Root Test is particularly useful in establishing the properties of Power Series. Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series Return to the Table of Content
69 SOLO Convergence Tests D’Alembert or Cauchy Ratio Test Jean Le Rond D’Alembert 1717 - 1783 If (an+1/an) ≤ r < 1 for all sufficiently large n, with r independent of n, then is convergent. If (an+1/an) ≥ 1 for all sufficiently large n, then is divergent. ∑n na ∑n na Convergence is proved by direct comparison with the geometric series (1+r+r2 + … )      = > < + ∞→ ateindetermin,1 ,1 ,1 lim 1 divergence econvergenc a a n n n ∑= = n i n n nS 1 2/Example: convergent n n a a n nn n n n 2 12 2 1 limlim 1 1 = + = +∞→ + ∞→ Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series Return to the Table of Content Augustin Louis Cauchy )1789-1857(
70 SOLO Convergence Tests Maclaurin or Euler Integral Test Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series Return to the Table of Content Let f (n) = an, i.e. f (x) is a monotonic decreasing function. Then converges if is finite and diverges if the integral is infinite. ∑n na ( )∫ ∞ 1 xdxf ( ) ( ) ( )1 111 fxdxfaxdxf n n +≤≤ ∫∑∫ ∞∞ = ∞ Colin Maclaurin 1698 - 1746 Leonhard Euler (1707 – 1`783) Is geometrically obvious that: ( )xf x 1 2 3 4 ( ) 11 af = ( ) 22 af = Comparison of Integral and Sum-Blocks Leading ( )xf x1 2 3 4 ( ) 11 af = Comparison of Integral and Sum-Blocks Lagging
SOLO Convergence Tests Kummer’s Test Consider a Series of positive terms ui and a sequence of positive constants ai. If for all n ≥ N, where N is some fixed number, then converges. If and diverges, then diverges. The two tests can be written as: Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series ∑ ∞ =1i iu 01 1 >≥− + + Ca u u a n n n n 01 1 ≤− + + n n n n a u u a ∑ ∞ =1i iu∑ ∞ = − 1 1 i ia    ∞→< > =      − −+ + ∞→ divergea converge Ca u u a i n n n n n 11 1 &0 0 lim Ernst Eduard Kummer (1810 – 1893)
72 SOLO Convergence Tests Kummer’s Test (continue – 1) Consider a Series of positive terms ui and a sequence of positive constants ai. If for all n ≥ N, where N is some fixed number, then converges. 01 1 >≥− + + Ca u u a n n n n ∑ ∞ =1i iu Ernst Eduard Kummer (1810 – 1893) nnnnn NNNNN NNNNN uauauC uauauC uauauC −≤ −≤ −≤ −− +++++ +++ 11 22112 111  Proof: Add and divide by C C ua C ua u nnNNn Ni i −≤∑ += 1 C ua u C ua C ua uuuuS NNN i i nnNNN i i n Ni i N i i n i in +<−+≤+== ∑∑∑∑∑ ==+=== 11111 The partial sums Sn have an upper bound. Since the lower bound is zero the sum must converge.∑ iu q.e.d. Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series
73 SOLO Convergence Tests Kummer’s Test (continue – 2) Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series Consider a Series of positive terms ui and a sequence of positive constants ai. If and diverges, then diverges. 01 1 ≤− + + n n n n a u u a ∑ ∞ =1i iu∑ ∞ = − 1 1 i ia Proof: Nnuauaua NNnnnn >≥≥≥ −− ,11  Since an > 0 n NN n a ua u ≥ and ∑∑ ∞ += − ∞ += ≥ 1 1 1 Ni iNN Ni i auau If diverges, then by comparison test diverges.∑ ∞ = − 1 1 i ia ∑ ∞ =1i iu Return to the Table of Content Ernst Eduard Kummer (1810 – 1893)
74 SOLO Convergence Tests Raabe’s Test If un > 0 and if for all n ≥ N, where N is a positive integer independent on, then converges. If Then diverges (as diverges). The limit form of Raabe’s test is Proof: In Kummer’s Test choose an = n and P = C + 1. Joseph Ludwig Raabe (1801 – 1859) Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series Return to the Table of Content 11 1 >≥      − + P u u n n n ∑i iu 11 1 ≤      − +n n u u n ∑i iu ∑ − i ia 1      = ∞→< > =      − ∑ − + ∞→ testno divergea converge P u u n i i n n n 1 &1 1 1lim 1 1
75 SOLO Convergence Tests Gauss’ s Test Carl Friedrich Gauss (1777 – 1855) If un > 0 for all finite n and in which B (n) is a bounded function of n for n → ∞, then converges for h > 1 and diverges for h ≤ 1. There is no indeterminate case here. ( ) 2 1 1 n nB n h u u n n ++= + ∑n nu Proof: For h > 1 and h < 1 the proof follows directly from Raabe’s Test: ( ) ( ) h n nB h n nB n h n u u n nn n n n =    +=    −++⋅=      −⋅ ∞→∞→ + ∞→ lim11lim1lim 2 1 If h = 1, Raabe’s Test fails. However if we return to Kummer’s Test and use an=n ln n: ( ) ( ) ( ) ( ) ( ) ( )             +−−+=    ++− + ⋅=       ++−    ++⋅ ∞→∞→ = ∞→ n nnnn n n nn nn n nB n h nn nn h n 1 1lnlnln1lim1ln1 1 lnlim 1ln11lnlim 1 2 Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series
76 SOLO Convergence Tests Gauss’ s Test Carl Friedrich Gauss (1777 – 1855) If un > 0 for all finite n and in which B (n) is a bounded function of n for n → ∞, then converges for h > 1 and diverges for h ≤ 1. There is no indeterminate case here. ( ) 2 1 1 n nB n h u u n n ++= + ∑n nu Proof (continue – 1): Kummer’s withan=n ln n: ( ) ( ) ( ) ( )             ++−=       ++−      ++⋅ ∞→ = ∞→ n nnn n nB n h nn n h n 1 1ln1lim1ln11lnlim 1 2 ( ) ( ) 01 3 1 2 11 1lim 1 1ln1lim 32 <−=      −+−⋅+−=      +⋅+− ∞→∞→  nnn n n n nn Hence we have a divergence for h = 1. This is an example of a successful application of Kremmer’s Test in which Raabe’s Test failed. Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series Return to the Table of Content
77 SOLO Complex Variables Infinite Series, Taylor’s and Laurent Series Let {un} :=u1 (z), u2 (z),…,un (z),…, be a sequence of single-valued functions of z in some region of z plane. We call U (z) the limit of {un} ,if given any positive number ε we can find a number N (ε,z) such that and we write this:( ) ( ) ( )zNnzUzun ,εε >∀<− ( ) ( ) ( ) ( )zUzuorzUzu n nn n ∞→ ∞→ →=lim x y C R If a sequence converges for all values z in a region R, we call R the region of convergence of the sequence. A sequence that is not convergent at some point z is called divergent at z. Infinite Series of Functions
78 SOLO Complex Variables Infinite Series, Taylor’s and Laurent Series Infinite Series of Functions From the sequence of functions {un} let form a new sequence {Sn} defined by: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑= =+++= += = n i inn zuzuzuzuzS zuzuzS zuzS 1 21 212 11   If , the series is called convergent and S (z) is its sum.( ) ( )zSzSnn =∞→ lim A necessary (but not sufficient) condition for convergence is that lim n→∞ un(z) = 0 Example: The Harmonic Series  ++++++=∑ ∞ = nnn 1 4 1 3 1 2 1 1 1 1 0 1 limlim == ∞→∞→ n u n n n By grouping the terms in the sum as ∞→+      + ++ + + + ++      ++++      +++ =>>>         2 1 22 1 2 1 1 2 1 1 1 8 1 7 1 6 1 5 1 4 1 3 1 2 1 1 p p pppp Return to the Table of Content
79 SOLO Complex Variables Absolute Convergence of Series of Functions Given a series of functions: ( ) ( )∑= = n i in zuzS 1 If is convergent the series is called absolutely convergent.( )∑= n i i zu 1 If is convergent but is not, the series is called conditionally convergent. ( )∑= n i i zu 1 ( )∑= n i i zu 1 Return to the Table of Content
80 SOLO Complex Variables Uniformly Convergence of Sequences and Series If for the sequence of functions {un(z)} we can find for each ε>0 a number N (ε) such that for all z∈R we say that {un} uniformly converges to U (z). ( N is a function only of ε and not of z) ( ) ( ) ( )εε NnzUzun >∀<− If the series of functions {Sn(z)} converges to S (z) for all z∈R we define the remainder ( ) ( ) ( ) ( )∑ ∞ += =−= 1 : nz inn zuzSzSzR The series of functions {Sn(z)} is uniformly convergent to S (z) if for all for all ε>0 and for all z∈R we can find a number N (ε) such that ( ) ( ) ( )εε NnzSzSn >∀<− x y C R Return to the Table of Content
81 SOLO Complex Variables Weierstrass M (Majorant) Test Karl Theodor Wilhelm Weierstrass (1815 – 11897) The most commonly encountered test for Uniform Convergence is the Weierstrass M Test. Proof: Since converges, some number N exists such that for n + 1 ≥ N, If we can construct a series of numbers , in which Mi ≥ |ui(x)| for all x in the interval [a,b] and is convergent, the series ui(x) will be uniformly convergent in [a,b]. ∑ ∞ 1 iM ∑ ∞ 1 iM ∑ ∞ 1 iM ε<∑ ∞ += 1ni iM This follows from our definition of convergence. Then, with |ui(x)| ≤ Mi for all x in the interval a ≤ x ≤ b, ( ) ε<∑ ∞ += 1ni i xu Hence ( ) ( ) ( ) ε<=− ∑ ∞ += 1ni in xuxsxS and by definition is uniformly convergent in [a,b]. Since we specified absolute values in the statement of the Weierstrass M Test, the series is also Absolutely Convergent. Return to the Table of Content ∑ ∞ =1i iu ∑ ∞ =1i iu
SOLO Complex Variables Abel’s Test Niels Henrik Abel ( 1802 – 1829) If and the functions fn(x) are monotonic decreasing | fn+1(x) ≤ fn(x)| and bounded, 0 ≤ fn(x) ≤ M, for all x in [a,b], then Converges Uniformly in [a,b]. ( ) ( ) convergentAa xfaxu n nnn ,= = ∑ ( )∑n n xu ( ) ( ) [ ] ( ) [ ]bainconvergentuniformlyisxu xd d baincontinuousarexu xd d andxu n nnn ,&, 1∑ ∞ = Return to the Table of Content Uniformly convergent series have three particular useful properties: 1.If the individual terms un(x) are continuous, the series sum is also continuous. 2. If the individual terms un(x) are continuous, the series may be integrated term by term. The sum of the integrals is equal to to the integral of the sum, then 3.The derivative of the series sum f (x) equals the sum of the individual term derivatives providing the following conditions are satisfied ( ) ( )∑ ∞ = = 1n n xuxf ( ) ( )∑ ∫∫ ∞ = = 1n b a n b a xdxuxdxf ( ) ( )∑ ∞ = = 1n n xu xd d xf xd d
SOLO Complex Variables Uniformly Convergent Series of Analytic Functions Suppose that (i)Each number of a sequence of functions u1(z), u2(z),…,un(z),… is Analytic inside a Region D, (ii)The Series is Uniformly Convergent through Every Region D’ interior to D. Then the function is Analytic inside D, and all its Derivatives can be calculated by term-by-term Differentiation. ( )∑ ∞ =1n n zu ( ) ( )∑ ∞ = = 1n n zuzf Proof: Let C be a simple closed contour entirely inside D, and let z a Point inside D. Since un(z) is Analytic inside D, we have: ( ) ( ) ∫ − = C n n wd zw wu i zu π2 1 for each function un(z). Hence ( ) ( ) ( ) ∑ ∫∑ ∞ = ∞ = − == 11 2 1 n C n n n wd zw wu i zuzf π
SOLO Complex Variables Uniformly Convergent Series of Analytic Functions Proof (continue – 1): Since is Uniformly Convergent on C, we may multiply by 1/(w-z) and integrate term-by-term: and we obtain ( ) ( ) ( ) ∑ ∫∑ ∞ = ∞ = − == 11 2 1 n C n n n wd zw wu i zuzf π ( )∑ ∞ =1n n zu ( ) ( ) ∑ ∫∫∑ ∞ = ∞ = − = − 11 n C n C n n wd zw wu wd zw wu ( ) ( ) ( ) ∫∫∑ − = − = ∞ = CC n n wd zw wf i wd zw wu i zf ππ 2 1 2 1 1 The last integral proves that f(z) is Analytic inside C, and since C is an arbitrary closed contour inside D, f(z) is Analytic inside D.
SOLO Complex Variables Uniformly Convergent Series of Analytic Functions Proof (continue – 2): Since f(z) is Analytic in D, the same is true for f’(z), therefore we can write ( ) ( ) ( )∫ − = C wd zw wf i zf 2 2 1 ' π Therefore q.e.d. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑∑ ∫ ∫∑∫ ∞ = ∞ = ∞ = = − = − = − = 11 2 1 22 ' 2 1 2 1 2 1 ' n nn C n eConvergenc Uniform C n n C zuwd zw wu i zw wd wu i wd zw wf i zf π ππ Hence the Series can be Differentiate term-by-term
SOLO Complex Variables Uniformly Convergent Series of Analytic Functions Remarks on the above Theorem (i)The contrast between the conditions for term-by-term differentiation of Real Series, and of Series of Analytic Functions is that - In the case of Real Series we have to assume that the Differentiated Series is Uniformly Convergent. - In the case of Series of Analytic Series the Theorem proved that the Differentiated Series is Uniformly Convergent. (ii)If we merely assumed that the given Series is Uniformly Convergent on a certain Closed Curve C, we could prove as before that f(z) is Analytic at all points inside C. (iii) Even if we assume that each un(z) is Analytic on the Boundary of the Domain D, and the Series is Uniformly Convergent on the Boundary, we can not prove that f(z) is Analytic on the Boundary, or the Differentiated Series Converges on the Boundary. (iv) The Theorem may be stated as a Theorem on Sequences of Functions: If fn(z) is Analytic in D for each value of n, and tends to f(z) Uniformly in any Region interior to D, then f(z) is Analytic inside D, and fn’(z) tends to f’(z) Uniformly in any Region interior to D. Return to the Table of Content
87 SOLO Complex Variables Let f (z) be analytic at all points within a circle C0 with center at z0 and radius r0. Then at each point z inside C0: Taylor’s Series ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  +−++−+−+= n n zz n zf zz zf zzzfzfzf 0 02 0 0 000 !!2 '' ' Power Series Brook Taylor 1685 - 1731 a convergent power series for some |z-z0|<R (radius of convergence). C x y R z0 C0 C1 z z' r0 r1 r Proof: Start with the Cauchy’s Integral Formula: ( ) ( ) ∫ − = C zd zz zf i zf ' ' ' 2 1 π Use the identity: α α ααα α − +++++≡ − − 1 1 1 1 12 n n  for: ( ) ( )                     − − − − − +        − − +++ − − + − = − − − − = −+− = − − nn zz zz zz zzzz zz zz zz zz zz zzzzzzzzzz 0 0 0 0 1 0 0 0 0 0 0 0000 ' ' 1 1 '' 1 ' 1 ' 1 1 ' 1 ' 1 ' 1  Since z inside C0 |z-z0|=r < r0. For z’ is on C1 we have |z’-z0|=r1<r0
88 SOLO Complex Variables Taylor’s Series (continue - 1) Power Series C x y R z0 C0 C1 z z' r0 r1 r Proof (continue - 1): Using the Cauchy’s Integral Formula: ( ) ( ) ∫ − = C zd zz zf i zf ' ' ' 2 1 π ( ) ( )                     − − − − − +        − − +++ − − + − = − − nn zz zz zz zzzz zz zz zz zz zf zz zf 0 0 0 0 1 0 0 0 0 0 ' ' 1 1 '' 1 ' ' ' '  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n R C n n n nzf C n zf C zf C Rzz n zf zzzfzf zzzz zdzf i zz zz zz zdzf i zz zz zdzf izz zdzf i n n +−++−+= −− − + −         − ++−         − + − = ∫ ∫∫∫ − 0 0 000 0 0 1 0 !/ 0 0 !1/' 2 00 ! ' '' '' 2 ' '' 2 1 ' '' 2 1 ' '' 2 1 0 0 0 0 0 0 0              π πππ We have: ( ) ( ) n i n n C n n n r r rr Mr der rrr Mr zzzz zdzfzz R       − = − ≤ −− − ≤ ∫∫ 11 1 2 0 1 110 0 2'' '' 2 0 π θ θ ππ where |f (z)|<M in C0 and r/r1< 1, therefore: 0 ∞→ → n nR q.e.d. 0100 ' rrzzrzz <=−<=−
89 SOLO Complex Variables Let f (z) be analytic at all points within a circle C0 with center at z0 and radius r0. Then at each point z inside C0: Taylor’s Series (continue – 2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  +−++−+−+= n n zz n zf zz zf zzzfzfzf 0 02 0 0 000 !!2 '' ' Power Series Brook Taylor 1685 - 1731 a convergent power series for some |z-z0|<R (radius of convergence). C x y R z0 C0 C1 z z' r0 r1 r Proof (continue – 2): ( ) ( ) ( ) ( )∑ ∞ = −= 0 01 0 1 !k k k zz k zf zfSuppose the series converges for z=z1: ( ) ( ) ( ) ( ) ( )         − − − =≤         − − −=−≤ <∞ = ∞ = ∞ = ∑∑∑ 01 0 1 001 0 0 01 0 0 0 0 1 !! zz zz M aM zz zz zz k zf zz k zf zf a k k k k k k k k k Since the series converges all its terms are bounded ( ) ( ) ,2,1,0 ! 01 0 =∀<− nMzz k zf k k Define: 01 0 : zz zz a − − = Therefore the series f (z) converges for all 010 zzzz −<− The region of convergence of a Taylor series of f (z) around a point z0 is a circle centered at z and radius of convergence R that extends until f (z) stops to be analytic.
90 SOLO Complex Variables Taylor’s Series (continue – 3) ( ) ( ) ( ) ( ) ( ) ( )  +++++= n n z n f z f zffzf ! 0 !2 0'' 0'0 2 Power Series Brook Taylor 1685 - 1731 When z0 = 0 the series is called Maclaurin’s series after Colin Maclaurin a contemporary of Brook Taylor. Colin Maclaurin 1698 - 1746 Examples of Taylor’s Series ∞<= ∑ ∞ = z n z e n n z 0 ! ( ) ( ) ∞< − −= ∑ ∞ = − + z n z z n n n 0 12 1 !12 1sin ( ) ( ) ∞<−= ∑ ∞ = z n z z n n n 0 2 !2 1cos ( ) ∞< − = ∑ ∞ = − z n z z n n 0 12 !12 sinh ( ) ∞<= ∑ ∞ = z n z z n n 0 2 !2 cosh ( ) 11 1 1 0 <−= + ∑ ∞ = zz z n nn Return to the Table of Content
91 SOLO Complex Variables Laurent’s Series (1843) Power Series If f (z) is analytic inside and on the boundary of the ring shaped region R bounded by two concentric circles C1 and C2 with center at z0 and respective radii r1 and r2 (r1 > r2), then for all z in R: Pierre Alphonse Laurent 1813 - 1854 C1 x y R C2R2 R1 z0 z z' r P1 P0 z'( ) ( ) ( )∑∑ ∞ = − ∞ = − +−= 1 00 0 n n n n n n zz a zzazf ( ) ( ) ,2,1,0' ' ' 2 1 2 1 0 = − = ∫ +−− nzd zz zf i a C nn π ( ) ( ) ,2,1,0' ' ' 2 1 1 1 0 = − = ∫ + nzd zz zf i a C nn π Proof: Since z is inside R we have R1 <|z-z0|=r < R2 , and |z’-z0|= R1 on C1 and R2 on C2. Start with the Cauchy’s Integral Formula: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫∫∫∫∫∫ − − − =→ − + − + − + − = 212 0 1 1 01 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 0 CCC P P P PC dz zz zf dz zz zf zfdzdz zz zf dz zz zf dz zz zf dz zz zf zf   
92 SOLO Complex Variables Laurent’s Series (continue - 1) Power Series Pierre Alphonse Laurent 1813 - 1854 C1 x y R C2R2 R1 z0 z z' r Proof (continue – 1): Since z and z’ are inside R we have R1 >|z-z0|=r >R2, |z’-z0|=R1. From Cauchy’s Integral Formula: ( ) ( ) ( ) ∫∫ − − − = 21 ' ' ' ' ' ' CC dz zz zf dz zz zf zf Use the identity: α α ααα α − +++++≡ − − 1 1 1 1 12 n n  For I integral:                     − − − − − +        − − ++ − − + − = − − − − = − − nn zz zz zz zzzz zz zz zz zz zz zzzzzz 0 0 0 0 1 0 0 0 0 0 0 00 ' ' 1 1 '' 1 ' 1 ' 1 1 ' 1 ' 1  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n R C n n n zs C n za C za C Rzzzazzzaza zzzz zdzf i zz zz zz zdzf i zz zz zdzf izz zdzf i n n +−⋅++−⋅+= −− − + −         − ++−         − + − = ∫ ∫∫∫ − 0000100 0 0 1 0 0 02 00 2 0 2 01 2 00 2 '' '' 2 ' '' 2 1 ' '' 2 1 ' '' 2 1              π πππ ( ) ∫ −1 ' ' ' 2 1 C zd zz zf iπ We have: ( ) ( ) n n n C n n n R r rR MR dR rRR Mr zzzz zdzfzz R       − = − ≤ −− − ≤ ∫∫ 11 1 2 0 1 110 0 2'' '' 2 0 π θ ππ where |f (z)|<M in R and r/R1< 1, therefore: 0 ∞→ → n nR
93 SOLO Complex Variables Laurent’s Series (continue - 2) Power Series Pierre Alphonse Laurent 1813 - 1854 C1 x y R C2R2 R1 z0 z z' r Proof (continue – 1): Since z and z’ are inside R we have R1 >|z-z0|=r > R2, |z’-z0|=R2. From Cauchy’s Integral Formula: ( ) ( ) ( ) ∫∫ − − − = 21 ' ' ' ' ' ' CC dz zz zf dz zz zf zf Use the identity: α α ααα α − +++++≡ − − 1 1 1 1 12 n n  For II integral:                     − − − − − +        − − ++ − − + − = − − − − = − − − nn zz zz zz zzzz zz zz zz zz zz zzzzzz 0 0 0 0 1 0 0 0 0 0 0 00 ' ' 1 1'' 1 1 ' 1 11 ' 1  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n R C n n n za C n za CC Rzzzazzza zzzz zdzfzz i zzzz zdzf izzzz zdzf i zdzf i n n − +− +− − − +−+−− +−++−= −− − + −        − ++ −        − += − +−− ∫ ∫∫∫ 1 001 1 001 0 0 1 0 1 00 2 0 0 0 01 0 01 00 ' ''' 2 1 1 ' '' 2 11 ' '' 2 1 '' 2 1              π πππ ( ) ∫ −C zd zz zf i ' ' ' 2 1 π We have: ( ) ( ) n n n C n n n r R rR RM dR rRr MR zzzz zdzfzz R       − = − ≤ −− − ≤ ∫∫− 2 2 2 2 0 2 2 2 0 0 2' ''' 2 1 0 π θ ππ where |f (z)|<M in R and R2/r< 1, therefore: 0 ∞→ − → n nR Return to the Table of Content
94 SOLO Complex Variables Zeros of Holomorphic Functions ( ) ( ) ( ) ( ) ( ) ( ) ( ) 00 00 1 0 1 0 ≠==== − zfandzfzfzf kk  We say that Holomorphic Function f (z) has a Zero of Order k at z = z0 if If f (z) has a Zero of Order k at z = z0, by Taylor expansion, we can write with Holomorphic and nonzero. ( ) ( ) ( )zfzzzf k k 0−= ( ) ( ) ( )kk zz zf zf 0 : − = Note: (1) For g (z) = 1/ f (z) the Order k Zeros of f (z) are Order k Poles of g (z) ( ) ( ) ( ) ( )zfzzzf zg k k 111 0− == (2) For ( ) ( ) ( ) ( ) ( )zf zf zz k zf zf zf zd d k k '' ln 0 + − == z = z0, is a Simple Pole. Return to the Table of Content
95 SOLO Complex Variables Theorem: f(z) Analytic and Nonzero → ln|f(z)| Harmonic If f (z) is analytic for in an Open Set Ω, and has no zeros in Ω, then ln |f(z)| is Harmonic in Ω. Proof : Since f (z) is analytic and has no zeros the logarithm of f(z) is also Analytic g (z) := ln f (z) is Analytic Therefore q.e.d. ( ) ( ) ( ) ( ) ( ) ( )zgizgzgizgzg eeeezf ImReImRe === + and ( ) Harmoniczgizgzg )(Im)(Re += ( ) ( ) ( ) ( )zgzgizg eeezf Re 1 ImRe =⋅=  ( ) ( ) ( ) Harmoniczgezf zg Relnln Re == meaning Harmonicyixgyixg yixg y yixg x yixg y yixg x )(Re),(Re 0)(Im)(Im&0)(Re)(Re 2 2 2 2 2 2 2 2 ++⇔ =+ ∂ ∂ ++ ∂ ∂ =+ ∂ ∂ ++ ∂ ∂ Return to the Table of Content
96 SOLO Complex Variables Polynomial Theorem If f (z) is analytic for all finite values of z, and as |z| → ∞, and then f (z) is a polynomial of degree ≤ k. Proof : Integrating this result we obtain q.e.d. ( ) kgivenAsomeforzforzAzf k &0, >∞→≤ Using Taylor Series around any analytic point z = a ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  +⋅ − ++⋅−+= af n az afazafzf n n ! 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Aafaz z az n af z az k af z az z af z zf nkn k k k k k kkk ≤+⋅−⋅ − ++⋅ − ++⋅ − +≤ −  ! 1 ! 11 If |z| → ∞ , the previous equation is possible only if f(n) (a)=0 for all n > k and all a. Therefore f (k) (a) = constant for all a, i.e. f (k) (z)=ak=constant. ( ) 0 1 1 azazazf k k k k +++= − −  Continuing to Integrate we obtain ( ) ( ) 1 1 − − += kk k azazf Return to the Table of Content
97 SOLO Complex Variables The Argument Theorem If f (z) is analytic inside and on a simple closed curve C except for a finite number of poles inside C (this is called a Meromorphic Function), then ( ) ( ) PNdz zf zf i C −=∫ ' 2 1 π where N and P are respectively the number of zeros and poles inside C. Proof: Let write f (z) as: ( ) ( ) ( ) ( )zG z z zf j p j k n k j k ∏ ∏ − − = α β where: ,∑∑ == j j k k pPnN & ( ) ( ) ( ) ( )zGzpznzf jjk k k lnlnlnln +−−−= ∑∑ αβ Differentiate this equation: ( ) ( ) ( ) ( ) ( ) ( ) Cinanalytic j j k k k zG zG z p z n zf zf ' ' + − − − = ∑∑ αβ ( ) ( ) ( ) ( ) ( ) ( ) PNpn zG zG iz p iz n i dz zf zf i j k k C p C j j k n C k k C jk −=−= + − − − = ∑∑ ∫∑∫∑ ∫∫        0 ' 2 1 2 1 2 1' 2 1 παπβππ and G (z) ≠ 0 and analytic in C (G’ (z) exists). ( ) ( ) k C k k C k k ndz z n i dz z n i k = − = − ∫∫ β βπβπ 2 1 2 1 ( ) ( ) j C j j C j j pdz z p i dz z p i j = − = − ∫∫ α απαπ 2 1 2 1 x y C R 1α 3 α 2α kα 1β 2β j β k Cα 3αC 2αC 2βC 1βC j Cβ q.e.d. Return to the Table of Content
98 SOLO Complex Variables Rouché’s Theorem Eugène Rouché 1832 - 1910 If f (z) and g (z) are analytic inside and on a simple closed curve C and if |g (z)| < |f (z)| on C, then f (z) + g (z) and f (z) have the same number of zeros inside C. Proof: Let F (z):= g (z)/f (z) If N1 and N2 are the number of zeros inside C of f (z) + g (z) and f (z) respectively, and using the fact that those functions are analytic and C, therefore they have no poles inside C, using the Argument Theorem we have ( ) ( )∫= C dz zf zf i N ' 2 1 2 π ( ) ( ) ( ) ( )∫ + + = C dz zgzf zgzf i N '' 2 1 1 π ( ) ( ) ( ) 0 1' 2 1 1 ' 2 1' 2 1 1 '' 2 1 ' 2 1 1 '1' 2 1' 2 1''' 2 1 32 21 = +−+−= + =−      + += − + ++ =− + ++ =− ∫∫∫∫ ∫∫∫∫ CCCC CCCC dzFFFF i dz F F i dz f f i dz F F f f i dz f f i dz Ff FfFf i dz f f i dz Fff FfFff i NN  ππππ ππππ We used the fact that |F|=|g/f|<1 on C, so the series 1-F+F2 +… is uniformly convergent on C and integration term by term yields the value zero. Thus N1=N2 q.e.d. Return to the Table of Content
99 SOLO Complex Variables Foundamental Theorem of Algebra (using Rouché’s Theorem) Every polynomial equation P (z) = a0 + a1z+a2z2 +…+anzn =0 with degree n ≥ 1 and an ≠ 0 (ai are complex constants) has at exactely n zeros. Proof: Define: Take C as the circle with the center at the origin and radius r > 1. q.e.d. ( ) n n zazf =: ( ) 1 1 2 210 : − − ++++= n n zazazaazg  ( ) ( ) ra aaaa ra rararara ra rararaa za zazazaa zf zg n n n n n n nnn n n n n n n n n 1210 1 1 1 2 1 1 1 0 1 1 2 210 1 1 2 210 − − − −−− − − − − ++++ = ++++ ≤ ++++ ≤ ++++ =   By choosing r large enough we can make |g (z)|/|f (z)|<1, and using Rouché’s Theorem ( ) ( ) ( ) n n n n zazazazaazgzfzP +++++=+= − − 1 1 2 210  and (n zeros at the origin z = 0) have the same number of zeros, i.e. P (z) has exactly n zeros. ( ) n n zazf =: Return to the Table of Content
100 SOLO Complex Variables Jensen’s Formula Johan Ludwig William Valdemar Jensen (1859 – 5 1925) which is the Mean-Value Property of the Harmonic Function ln |f(z)|. Suppose that ƒ is an Analytic Function in a region in the complex plane which contains the closed disk D of radius r about the origin, a1, a2, ..., an are the zeros of ƒ in the interior of D repeated according to multiplicity, and ƒ(0) ≠ 0. Jensen's formula states that This formula establishes a connection between the Moduli of the zeros of the function ƒ(z) inside the disk D and the average of log |f(z)| on the boundary circle |z| = r, and can be seen as a generalization of the Mean Value Property of Harmonic Functions. Namely, if f(z) has no zeros in D, then Jensen's formula reduces to ( ) ( )∫∑ +        = = π θ θ π 2 01 ln 2 1 ln0ln derf r a f j n k k ( ) ( )∫= π θ θ π 2 0 ln 2 1 0ln derff j
101 SOLO Complex Variables Jensen’s Formula (continue – 1) Proof If f has no Zeros in D, then we can use Gauss’ Main Value Theorem to ln f(z) that is Harmonic in D ( ) ( )∫= π θ θ π 2 0 ln 2 1 0ln derff j Since f has Zeros a1, a2,…, an inside D, ( |z| < r) let define the Holomorphic Function F (z) without Zeros in D : ( ) ( ) ( )∏= − − ⋅= n k k k raz rza zfzF 1 2 / /1 : Apply the Gauss’ Main Value theorem for |z|=r ( ) ( ) ( ) ( )∫∫∑ ==        −= = π θ π θ θ π θ π 2 0 2 01 ln 2 1 ln 2 1 ln0ln0ln derfderF r a fF ii n k k and:   1 1 1 1 /1 1 /1 / /1 / /1 2 2 1 2 22 1 22 22 = − − = − − = − − = − − = − − →=⋅= r z a r z a r zz z a rza z a rza z r raz rza raz rza rzzz k k k k k k k k k k The Zeros of f(z) are cancelled. q.e.d. We have: i.e. zk1 is outside the Disk D. rzarzforrza k ra kkkk k >→==− < 1 2 1 2 1 /0/1
102 SOLO Complex Variables Jensen’s Formula (continue – 2) Johan Ludwig William Valdemar Jensen (1859 – 5 1925) ( ) ( ) ( )zh zg zzf l = Jensen's formula may be generalized for functions which are merely meromorphic on D. Namely, assume that where g and h are analytic functions in D having zeros at respectively, then Jensen's formula for meromorphic functions states that Jensen's formula can be used to estimate the number of zeros of analytic function in a circle. Namely, if f is a function analytic in a disk of radius R centered at z0 and if |f(z)| is bounded by M on the boundary of that disk, then the number of zeros of f(z) in a circle of radius r < R centered at the same point z0 does not exceed { } { }0,,0,, 11 DbbandDaa mn ∈∈  ( ) ( ) ( )∫+= − π θ π 2 01 1 ln 2 1 ln 0 0 ln i m nnm erf bb aa r h g   ( ) ( )0 ln /ln 1 zf M rR
103 SOLO Complex Variables Jensen’s Formula (continue – 3) Johan Ludwig William Valdemar Jensen (1859 – 5 1925) Jensen's formula may be put in an other way. If n (r) denotes the number of Zeros, including multiplicity, and p (r) denotes the number of Poles, including multiplicity, for |z| < r, then the Jensen’s Formula can be written as ( ) ( ) ( ) ( )0lnln 2 1 lnln 2 0110 fderf b r a r dx x xpxn j n j j m k k r −=         −        = − ∫∑∑∫ == π θ θ π Proof         −−−=         −        ∑∑∑∑ ==== n j j m k k n j j m k k brnarm b r a r 1111 lnlnlnlnlnln ( ) ( ) ( ) ( )        −+−−−+−= ∑∑ − = + − = + n n j jjm m k kk brnbbjarnaak lnlnlnlnlnlnlnln 1 1 1 1 1 1         +−         += ∫∑ ∫∫∑ ∫ − = − = ++ r r n j r r r r m k r r n j jn k k x xd n x xd j x xd m x xd k 1 1 1 1 11
104 SOLO Complex Variables Jensen’s Formula (continue – 4) Johan Ludwig William Valdemar Jensen (1859 – 5 1925) Jensen's formula may be put in an other way. If n (r) denotes the number of Zeros, including multiplicity, and p (r) denotes the number of Poles, including multiplicity, for |z| < r, then the Jensen’s Formula can be written as ( ) ( ) ( ) ( )0lnln 2 1 lnln 2 0110 fderf b r a r dx x xpxn j n j j m k k r −=         −        = − ∫∑∑∫ == π θ θ π Proof (continue – 1)         +−         +=         −        ∫∑ ∫∫∑ ∫∑∑ − = − === ++ r r n j r r r r m k r r n j j m k k n j jn k k x xd n x xd j x xd m x xd k b r a r 1 1 1 111 11 lnln But k = n (x) for rm ≤ x ≤ rm+1, m = n (x) for rn ≤ x ≤ r, and j = p (x) for rj ≤ x ≤ rj+1, n = p (x) for rn ≤ x ≤ r Hence ( ) ( ) xd x xp xd x xn b r a r rrn j j m k k ∫∫∑∑ −=         −        == 0011 lnln q.e.d. Return to the Table of Content
105 SOLO Complex Variables Poisson-Jensen’s Formula for a Disk Johan Ludwig William Valdemar Jensen (1859 – 5 1925) Poisson Formula states: Siméon Denis Poisson 1781-1840 Let g (z) be analytic inside and on the circle C defined by |z| = R, and let z = r e iθ be any point inside C, then: ( ) ( )∫ − − = π φ θφ θ φ π 2 0 2 22 2 1 deRg ereR rR erg i ii i In our case ƒ is an analytic function in a region in the complex plane which contains the closed disk D of radius R about the origin, a1, a2, ..., am are the Zeros, and b1, b2, ..., bn are the Poles of ƒ in the interior of D repeated according to multiplicity. Since f has Zeros a1, a2,…, an inside D, ( |z| < r) let define the Holomorphic Function F (z) without Zeros in D : ( ) ( ) ( ) ( ) Rz Rzb Rbz Raz Rza zfzF m k n j j j k k = − − ⋅ − − ⋅= ∏ ∏= =1 1 2 2 /1 / / /1 : Zeros and Poles of f(z) are cancelled, and new ones are outside D.
106 SOLO Complex Variables Poisson-Jensen’s Formula for a Disk Johan Ludwig William Valdemar Jensen (1859 – 5 1925) Let apply the Poisson Formula to g (z) = ln |F (z)|: ( ) ( ) ( ) ( ) ( ) ∏∏ == − − + − − +== n j j j m k k k Rzb Rbz Raz Rza zfzFzg 1 2 1 2 /1 / ln / /1 lnlnln: ( ) ( ) 1 /1 / / /1 2 2 Rz j j Rz k k Rzb Rbz Raz Rza == = − − = − − Siméon Denis Poisson 1781-1840 We proved that: ( ) ( )φφ ii eRfeRg ln= The Poisson Formula to g (z) = ln F (z) is: ( ) ( ) ( ) ( ) RzdeRf ereR rR Rzb Rbz Raz Rza zf i ii n j j j m k k k < − − = − − + − − + ∫ ∏∏ == π φ θφ φ π 2 0 2 22 1 2 1 2 ln 2 1 /1 / ln / /1 lnln RzRzbandRzRza jjjkkk >→=−>→=− 1 2 11 2 1 0/1,0/1
107 SOLO Complex Variables Poisson-Jensen’s Formula for a Disk Johan Ludwig William Valdemar Jensen (1859 – 5 1925) Siméon Denis Poisson 1781-1840 The Poisson-Jensen’s Formula for a Disk is: ( ) ( ) ( ) ( ) RzdeRf ereR rR Rzb Rbz Raz Rza zf i ii n j j j m k k k < − − = − − + − − + ∫ ∑∑ == π φ θφ φ π 2 0 2 22 1 2 1 2 ln 2 1 /1 / ln / /1 lnln For z = r = 0 we obtain the Jensen’s Formula: ( ) ( )∫=       + − π φ φ π 2 021 21 ln 2 1 ln0ln deRfR aaa bbb f inm m n   If there are no Zeros or Poles in D, it reduces to Poisson’s Formula: ( ) ( ) RzdeRf ereR rR zf i ii < − − = ∫ π φ θφ φ π 2 0 2 22 ln 2 1 ln Return to the Table of Content
108 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series If f (z) is analytic inside and on the boundary of a circle C, except it’s center z0, then according to Laurent’s Series: C x y R R z0 z z' r P0 ( ) ( ) ( )∑∑ ∞ = − ∞ = − +−= 1 00 0 n n n n n n zz a zzazf ( ) ( ) ,2,1,0' ' ' 2 1 1 0 = − = ∫ +−− nzd zz zf i a C nn π ( ) ( ) ,2,1,0' ' ' 2 1 1 0 = − = ∫ + nzd zz zf i a C nn π Let compute ( ) ( ) ( ) ( ) ,2,1,0' ' ' '''' 1 00 0 = − +−= ∑ ∫∫ ∑ ∫ ∞ = − ∞ = nzd zz zf azdzzazdzf n C nn C n C n n ( ) ( ) ( )    = ≠ = − ==− ∫∫ 12 10 ' ' ' &,2,1,00'' 0 0 ni n zd zz zf nzdzz C n C n π  Therefore: ( ) 12'' −∫ = aizdzf C π Because only a-1 is involved in the integral above, it is called the residue of f (z) at z = z0. Return to the Table of Content
109 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series According to residue definition the residue of f (z) at z = z0 can be computed as follows: C x y R R z0 z z' r P0 If z = z0 is a pole of order k, i.e. the Laurent series at z0 is Then: ( )∫=− C zdzf i a '' 2 1 1 π Calculation of the Residues ( ) ( ) ( ) ( ) ( )  +−+−++ − ++ − + − = −−− 2 02010 0 2 0 2 0 1 zzazzaa zz a zz a zz a k k ( ) ( ) ( )[ ]zfzz zd d k a k k k zz 01 1 1 !1 1 lim 0 − − = − − →− ( ) ( ) ( ) ( ) ( ) ( ) ( )  +−+−+−+++−+−=− ++ − − − − − 2 02 1 0100 2 02 1 010 kkk k kkk zzazzazzaazzazzazfzz and: If z = z0 is a pole of order k=1, then: ( ) ( )[ ]zfzza zz 01 0 lim −= →− ( ) ( ) ( )∑∑ = − ∞ = − +−= k n n n n n n zz a zzazf 1 00 0 Return to the Table of Content
110 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series The Residue Theorem If f (z) is analytic inside and on the boundary of a closed curve C, except at the singularities z01, z02,…,z0n, which have residues Re1, Re2,…,Ren, then: Proof: ( ) ( )n C izdzf ReReRe2'' 21 +++=∫ π x y C R 01z nzC 2zC 02z 1zC nz0 Surround every singularity z0i by a small closed curve Czi, that enclosed only this singularity. Connect those Curves to C by a small corridor (the width of which shrinks to zero, so that the integration along the opposite directions will cancel out) ( ) ( ) ( ) ( ) 0'''''''' 21 =−−−− ∫∫∫∫ znzz CCCC zdzfzdzfzdzfzdzf  We have , therefore:( ) i C izdzf zi Re2'' π∫ = ( ) ( ) ( ) ( ) ( )n CCCC izdzfzdzfzdzfzdzf znzz ReReRe2'''''''' 21 21 +++=+++= ∫∫∫∫  π q.e.d. Return to the Table of Content
111 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Theorem 1 If |F (z)| ≤ M/Rk for z = R e iθ where k > 1 and M are constants, then where Γ is the semicircle arc of radius R, center at origin, in the upper part of z plane. ( ) 0lim =∫Γ →∞ zdzFR x y Γ R Proof: ( ) 1 0 1 0 == = ΓΓ =≤=≤ ∫∫∫∫ kk i k eRz k R M d R M deRi R M zd R M zdzF i π θθ ππ θ θ Therefore: ( ) 0limlim 1 1 0 1 > −→∞− Γ →∞ ==≤ ∫∫ k kRkR R M d R M zdzF π θ π ( ) 0lim =∫Γ →∞ zdzFR and: ( ) ( ) ( ) 0limlimlim0 =≤≤−= ∫∫∫ Γ →∞ Γ →∞ Γ →∞ zdzFzdzFzdzF RRR q.e.d.Return to the Table of Content
112 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Jordan’s Lemma If |F (z)| ≤ M/Rk for z = R e iθ where k > 0 and M are constants, then where Γ is the semicircle arc of radius R, center at origin, in the upper part of z plane, and m is a positive constant. ( ) 0lim =∫Γ →∞ zdzFe zmi R x y Γ R Proof: ( ) 0lim =∫Γ →∞ zdzFe zmi R using: q.e.d. ( ) ( )∫∫ = Γ = π θθ θ θ θ 0 deRieRFezdzFe iieRmi eRz zmi i i ( ) ( ) ( ) ( ) ∫∫∫ ∫∫∫ − − − − − − =≤= =≤ 2/ 0 sin 1 0 sin 1 0 sin 0 sincos 00 2 π θ π θ π θθ π θθθθ π θθ π θθ θθθ θθθ θθ dRe R M dRe R M dReRFe deRieRFedeRieRFedeRieRFe Rm k Rm k iRm iiRmRmiiieRmiiieRmi ii 2/0/2sin πθπθθ ≤≤≥ for π2/π 1 θsin πθ /2 θ ( ) ( )Rm k Rm k Rm k iieRmi e R M de R M de R M deRieRFe i −− − − − −=≤≤ ∫∫∫ 1 222 2/ 0 /2 1 2/ 0 sin 1 0 π π π θ π θθ θθθ θ ( ) ( ) 01 2 limlim 0 =−≤ − →∞→∞ ∫ Rm kR iieRmi R e R M deRieRFe i π θθ θ θ Marie Ennemond Camille Jordan 1838 - 1922
113 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Jordan’s Lemma Generalization If |F (z)| ≤ M/Rk for z = R e iθ where k > 0 and M are constants, then for Γ a semicircle arc of radius R, and center at origin: ( ) 00lim >=∫Γ →∞ mzdzFe zmi R x y Γ R where Γ is the semicircle, in the upper part of z plane. 1 ( ) 00lim <=∫Γ →∞ mzdzFe zmi R x y Γ R where Γ is the semicircle, in the down part of z plane. 2 ( ) 00lim >=∫Γ →∞ mzdzFe zm R x y Γ R where Γ is the semicircle, in the right part of z plane. 3 ( ) 00lim <=∫Γ →∞ mzdzFe zm R where Γ is the semicircle, in the left part of z plane. 4 x yΓ R Return to the Table of Content
114 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Integral of the Type Bromwwich-Wagner ( )∫ ∞+ ∞− jc jc ts sdsFe iπ2 1 The contour from c - i ∞ to c + i ∞ is called Bromwich Contour Thomas Bromwich 1875 - 1929 x y 0< Γt R c x y 0>Γt R c ( ) ( ) ( ) ( ) ( ) ( ) ( )    < > ==         +== ∫ ∫∫∫ Γ ∞+ ∞− →∞ ∞+ ∞− 0 0 2 1 lim 2 1 2 1 tzFeRes tzFeRes zdzF i sdsFesdsFe i sdsFe i tf tz planezRight tz planezLeft ts ic ic ts R ic ic ts π ππ where Γ is the semicircle, in the right part of z plane, for t < 0. where Γ is the semicircle, in the left part of z plane, for t > 0. This integral is also the Inverse Laplace Transform. Return to the Table of Content
115 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Integral of the Type ,F (sin θ, cos θ) is a rational function of sin θ and cos θ ( )∫ π θθθ 2 0 cos,sin dF Let z = e iθ 22 cos, 22 sin 11 −−−− + = + = − = − = zzee i zz i ee iiii θθθθ θθ zizdddzideizd i /=→== θθθθ ( ) ∫∫       +− = −− C zi zdzz i zz FdF 2 , 2 cos,sin 112 0 π θθθ where C is the unit circle with center at the origin. C x y R=1 Return to the Table of Content
116 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Definite Integrals of the Type .( )∫ +∞ ∞− xdxF If the conditions of Theorem 1, i.e.: if |F (z)| ≤ M/Rk for z = R e iθ where k > 1 and M are constants, then and we can write ( ) 0lim =∫Γ →∞ zdzFR x y Γ R ( ) ( ) ( ) ( ) ( )zFResizdzFzdzFxdxFxdxF planezUpper R R R π2lim ==         += ∫∫∫∫ Γ + − →∞ +∞ ∞− Example: Heaviside Step Function ( ) x e i xF txi π2 1 := x y Γ R 0>t x y Γ R 0<t This function has a single pole at z = 0. For t > 0 Γ is the semicircle, in the upper part of z plane. We also include on the path x = - ∞ to x = + ∞ a small semicircle such that the pole z = 0 is included. For t < 0 Γ is the semicircle, in the lower part of z plane. We also include on the path x = - ∞ to x = + ∞ a small semicircle such that the pole z = 0 is excluded.     <= >= =∫ +∞ ∞− 00/ 01/ 2 1 tzeRes tzeRes xd x e i tzi planezLower tzi planezUpper txi π Return to the Table of Content
117 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Cauchy’s Principal Value Cauchy’s Principal value deals with integrals that have singularities along the integration paths. Start with the following: Theorem 1: If f (z) is analytic on and inside a positive-sensed circle C of radius ε, centered at z = z0, then ( ) ( )0 0 0 lim zfi zz zdzf C ψ ψ ε = −∫→ where Cψ is every arc on C of angle ψ. Proof: Since f (z) is analytic inside and on C we can use the Taylor series expansion to write ( ) ( ) ( ) ( ) ( )∑ ∞ = −+= 1 0 0 0 !n n n zz n zf zfzf ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )    zg n n n zz n zf zz zf zz zf ∑ ∞ = − −+ − = − 1 1 0 0 0 0 0 ! Consider the integral on Cψ defined by z=z0 + ε e iθ θ0 ≤ θ ≤ θ0 + ψ C x y ψC 0 z 0 θ 0 θψ + ψ ε O
118 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Cauchy’s Principal Value (continue – 1) Proof (continue – 1): Since g (z) is bounded inside and on C , there is a positive number M such that |g (z)| < M for all z such that |z – z0| < ε ( ) ( ) ( )∫∫∫ + − = − ψψψ CCC zdzg zz zdzf zz zdzf 0 0 0 ( ) ( ) ( ) ( )00 0 0 0 0 0 0 0 zfidizf zz zd zf zz zdzf i ezz CC ψθ ψθ θ ε θ ψψ == − = − ∫∫∫ + += ( ) ( ) ( )εψ ψψ MLMzdzgzdzg CC =<≤ ∫∫ Where L = ψ ε is the length of Cψ. ( ) ( ) ( ) 0lim0limlim 000 =⇒=≤ ∫∫ →→→ ψψ εεε εψ CC zdzgMzdzg ( ) ( ) ( ) ( ) 0lim0limlimlim0 0000 =⇒=≤≤−= ∫∫∫∫ →→→→ ψψψψ εεεε CCCC zdzgzdzgzdzgzdzg Therefore ( ) ( )0 0 0 lim zfi zz zdzf C ψ ψ ε = −∫→ q.e.d. Note: For ψ = 2 π we recover the Cauchy’s Integral result. C x y ψC 0 z 0 θ 0θψ + ψ ε O
119 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Cauchy’s Principal Value (continue – 2) Theorem 2: If F (z) is analytic on and inside a positive-sensed circle C of radius ε, except at the center of C, z = z0, that is a simple pole of F (z), then C x y ψC 0 z 0 θ 0 θψ + ψ ε O ( ) ( )[ ]0 0 lim zFResizdzF C ψ ψ ε =∫→ where Cψ is every arc on C of angle ψ. Proof: Since f (z) is analytic inside and on C by using Theorem 1 we obtain the desired result. The function: ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]    =−= ≠− = → 000 00 0 lim zzzFzzzFRes zzzFzz zf zz
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Mathematics and History of Complex Variables
Mathematics and History of Complex Variables
Mathematics and History of Complex Variables
Mathematics and History of Complex Variables
Mathematics and History of Complex Variables
Mathematics and History of Complex Variables
Mathematics and History of Complex Variables
Mathematics and History of Complex Variables
Mathematics and History of Complex Variables
Mathematics and History of Complex Variables
Mathematics and History of Complex Variables
Mathematics and History of Complex Variables
Mathematics and History of Complex Variables
Mathematics and History of Complex Variables
Mathematics and History of Complex Variables
Mathematics and History of Complex Variables

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Mathematics and History of Complex Variables

  • 1. 1 Complex Variables SOLO HERMELIN Updated: 11.05.07 28.10.12 http://www.solohermelin.com
  • 2. 2 SOLO Complex Variables Table of Contents Set of Numbers – Examples Fundamentals Operations with Complex Numbers z = x + i y Axiomatic Foundations of the Complex Number System ( ) R∈= babaz ,, History of Complex Numbers Derivatives Cauchy-Riemann Equations Harmonic Functions Orthogonal Families Singular Points Complex Line Integrals Simply and Multiply Connected Regions Green’s Theorem in the Plane Consequences of Green’s Theorem in the Plane Cauchy’s Theorem Cauchy-Goursat Theorem Consequences of Cauchy-Goursat Theorem
  • 3. SOLO Complex Variables Table of Contents (continue - 1) Cauchy’s Integral Formulas and Related Theorems Cauchy’s Integral Formulas Cauchy’s Integral Formulas for the n Derivative of a Function Morera’s Theorem (the converse of Cauchy’s theorem) Cauchy’s Inequality Liouville’s Theorem Foundamental Theorem of Algebra Gauss’ Mean Value Theorem Maximum Modulus Theorem Minimum Modulus Theorem Poisson’s Integral Formulas for a Circle Poisson’s Integral Formulas for a Half Plane
  • 4. 4 SOLO Complex Variables Table of Contents (continue - 2) Theorems of Convergence of Sequences and Series Convergence Tests Cauchy Root Test D’Alembert or Cauchy Ratio Test Maclaurin or Euler Integral Test Kummer’s Test Raabe’s Test Gauss’ s Test Infinite Series, Taylor’s and Laurent Series Infinite Series of Functions Absolute Convergence of Series of Functions Uniformly Convergence of Sequences and Series Weierstrass M (Majorant) Test Abel’s Test Uniformly Convergent Series of Analytic Functions Taylor’s Series Laurent’s Series (1843)
  • 5. 5 SOLO Complex Variables Table of Contents (continue - 3) 5 The Argument Theorem Rouché’s Theorem Foundamental Theorem of Algebra (using Rouché’s Theorem) Zeros of Holomorphic Functions Theorem: f(z) Analytic and Nonzero → ln|f(z)| Harmonic Polynomial Theorem Jensen’s Formula Poisson-Jensen’s Formula for a Disk
  • 6. 6 SOLO Complex Variables Table of Contents (continue - 4) Calculation of the Residues The Residue Theorem, Evaluations of Integral and Series The Residue Theorem Evaluation of Integrals Jordan’s Lemma Integral of the Type Bromwwich-Wagner Integral of the Type ,F (sin θ, cos θ) is a rational function of sin θ and cos θ ( )∫ π θθθ 2 0 cos,sin dF Definite Integrals of the Type .( )∫ +∞ ∞− xdxF Cauchy’s Principal Value Differentiation Under Integral Sign, Leibnitz’s Rule Summation of Series Infinite Products The Mittag-Leffler and Weierstrass , Hadamard Theorems The Weierstrass Factorization Theorem The Hadamard Factorization Theorem Mittag-Leffler’s Expansion Theorem Analytic Continuation Conformal Mapping
  • 7. 7 SOLO Complex Variables Douglas N. Arnold Gamma Function Bernoulli Numbers Fourier Transform Laplace Transform Z Transform Mellin Transform Hilbert Transform Zeta Function Table of Contents (continue - 5) Applications of Complex Analysis References
  • 8. 8 SOLO Algebra Set of Numbers – Examples { }+∞<<∞−= xnumberrealaisxx ,:R Set of real numbers { }+∞<<−∞−=+== yxiyixznumbercomplexaiszzC ,,1,,: Set of complex numbers { } ,3,2,1,0,1,2,3,,: −−−= integeranisiiZ Set of integers { },3,2,1,0,0: integernaturalaisnnN ≥= Set of positive integers or natural numbers { }0,,,/: ≠∈== qZqpwhereqprrQ Set of rational numbers We have: CZN ⊂⊂⊂ R { }QxxIR −∈= R: Set of irrational numbers ∅== IRQIRQ  &R
  • 9. 9 SOLO Complex Variables Complex numbers can result by solving algebraic equations a cabb x 2 42 1 −−− = a cabb x 2 42 2 −+− = 042 >−=∆ cab a b xx 2 21 − == 042 =−=∆ cab 042 <−=∆ caby x cxbxay ++= 2 a bcaib x 2 4 2 2,1 −+− = y x dxcxbxay +++= 23 Three real roots for y = 0 One real & two complex roots for y = 0 πki ez 25 1== y x 5 2 2 π i ez = 5 2 2 2 π i ez = 5 2 3 3 π i ez = 5 2 4 4 π i ez = 11 =z  72  72  72  72  72 1. Quadratic equations 2. Cubic equations 3. Equation Examples 02 =++ cxbxa 023 =+++ dxcxbxa 015 =−x Return to Table of Contents
  • 10. 10 SOLO Complex Variables Fundamentals Operations with Complex Numbers z = x + i y ( ) 1 ,sincos , 2 −=    ==+ ==+ = i ArgumentModulusi partImaginaryypartRealxyix z θρθθρ θ ρ i eyixz =+= y x ρ θ Division Addition ( ) ( ) ( ) ( )dbicadicbia +++=+++ Subtraction ( ) ( ) ( ) ( )dbicadicbia −+−=+−+ Multiplication ( ) ( ) ( ) ( )cbdaidbcadbicbidaicadicbia ++−=+++=++ 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )dccdidc dacbidbca dic dic dic bia dic bia −++ −++ = − − + + = + + 22 ( ) ( ) ( ) ( ) ( ) ( ) 022 2222 ≠+ + − + + + = + + dc dc dacb i dc dbca dic bia Conjugate ( )θθρ sincos:* iyixz −=−= Absolute Value ρ==+= *22 : zzyxz θ ρ i eyixz − =−=:* θ ρ i eyixz =+=: x y ρ ρ θ θ−
  • 11. 11 SOLO Complex Variables Fundamentals Operations with Complex Numbers z = x + i y θ ρ i eyixz =+= y x ρ θ Polar Form of a Complex Number Multiplication Division ( )2121 212121 θθθθ ρρρρ + =⋅=⋅ iii eeezz *22 zzyxz =+==ρ ( )θθρ sincos: iyixz +=+= ( )xy /tan 1− =θ ( )2121 212121 /// θθθθ ρρρρ − == iii eeezz Euler’s Formula ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )θθ θθθθθθ θθθ θθ θ sincos !12 1 !3!1!2 1 !4!2 1 !!2!1 1 sin 123 cos 242 2 i k i k n iii e k k k k n i +=             + + −++−++−+++−= +++++= +          Leonhard Euler 1707- 1783 ( ) 1 ,sincos , 2 −=    ==+ ==+ = i ArgumentModulusi partImaginaryypartRealxyix z θρθθρ
  • 12. 12 SOLO Complex Variables Fundamentals Operations with Complex Numbers z = x + i y 1 , , 2 −=    == ==+ = i ArgumentModuluse partImaginaryypartRealxyix z i θρρ θ θ ρ i eyixz =+= y x ρ θ Polar Form of a Complex Number θ ρ i eyixz =+=: De Moivre Theorem ( )[ ] ( ) ( )θθρρ ρθθρ θ θ nine eiz nnin ninn sincos sincos +== =+= Roots of a Complex Number [ ] ( ) ( )[ ] 1.2.1.0 2 sin 2 cos sincos /1 /1/12/1 −=            + +      + = +== + nk n k i n k iez n nnkin  πθπθ ρ θθρρ πθ πki ez 25 1== y x 5 2 2 π i ez = 5 2 2 2 π i ez = 5 2 3 3 π i ez = 5 2 4 4 π i ez = 11 =z  72  72  72  72  72 Abraham De Moivre 1667 - 1754 Return to Table of Contents
  • 13. 13 SOLO Complex Variables Axiomatic Foundations of the Complex Number System ( ) R∈= babaz ,, Definition of Complex System: From those relation, for any complex numbers z1,z2,z3 ∈ C we obtain: CzzCzzCzz ∈∀∈⋅∈+ 212121 ,& Closure Law1 1221 zzzz +=+ Commutative Law of Addition2 ( ) ( ) 312321 zzzzzz ++=++ Associative Law of Addition3 1221 zzzz ⋅=⋅ Commutative Law of Multiplication4 ( ) ( ) 312321 zzzzzz ⋅⋅=⋅⋅ Associative Law of Multiplication5 ( ) 3121321 zzzzzzz ⋅+⋅=+⋅ Distributive Law6 111 00 zzz =+=+ 111 11 zzz =⋅=⋅7 0.. 11 =+∈∃∈∀ zztsCzuniqueCz zz −=18 1..0 11 =⋅∈∃∈≠∀ zztsCzuniqueCz zzz /11 1 == − 9 Equality ( ) ( ) dbcadcba ==⇔= ,,,A Sum ( ) ( ) ( ) ( )dbcadcba +++=+ ,,B Product ( ) ( ) ( )cbdadbcadcba +−=⋅ ,,, ( ) ( ) R∈= mbmambam &,,C Return to Table of Contents
  • 14. 14 SOLO Complex Variables History of Complex Numbers Brahmagupta (598-670) writes Khandakhadyaka (665) which solves quadratic equations and allows for the possibility of negative solutions. Brahmagupta 598 - 670 Brahmagupta also solves quadratic equations of the type a x2 + c = y2 and a x2 - c = y2 . For example he solves 8x2 + 1 = y2 obtaining the solutions (x,y) = (1,3), (6,17), (35,99), (204,577), (1189,3363), ... For the equation 11x2 + 1 = y2 Brahmagupta obtained the solutions (x,y) = (3,10), (161/5,534/5), ... He also solves 61x2 + 1 = y2 which is particularly elegant having x = 226153980, y = 1766319049 as its smallest solution.
  • 15. 15 SOLO Complex Variables History of Complex Numbers Abraham bar Hiyya Ha-Nasi ‫הנשיא‬ ‫חייא‬ ‫בר‬ ‫אברהם‬ writes the work Hibbur ha-Meshihah ve-ha-Tishboret ‫והתשבורת‬ ‫המשיחה‬ ‫חבור‬ , translated in 1145 into Latin as Liber embadorum, which presents the first complete solution to the quadratic equation. Abraham bar Hiyya Ha-Nasi (‫הנשיא‬ ‫חייא‬ ‫בר‬ ‫אברהם‬ Abraham son of [Rabbi] Hiyya "the Prince") (1070 - 1136?) was a Spaish Jewish Mathematician and astronomer, also known as Savasorda (from the Arabic ‫الشرطة‬ ‫صاحب‬ Sâhib ash-Shurta "Chief of the Guard"). He lived in Barcelona. Abraham bar iyya ha-NasiḤ [2] (1070 – 1136 or 1145)
  • 16. 16 SOLO Complex Variables History of Complex Numbers Nicolas Chuquet (1445 – 1488) Chuquet wrote an important text Triparty en la science des nombres. This is the earliest French algebra book . The Triparty en la science des nombres (1484) covers arithmetic and algebra. It was not printed however until 1880 so was of little influence. The first part deals with arithmetic and includes work on fractions, progressions, perfect numbers, proportion etc. In this work negative numbers, used as coefficients, exponents and solutions, appear for the first time. Zero is used and his rules for arithmetical operations includes zero and negative numbers. He also uses x0 = 1 for any number x. The sections on equations cover quadratic equations where he discusses two solutions.
  • 17. 17 SOLO Complex Variables History of Complex Numbers Girolamo Cardano 1501 - 1576 Nicolo Fontana Tartaglia 1500 - 1557 Solution of cubic equation x3 + b x2 +c x +d = 0 The first person to solve the cubic equation x3 +b x = c was Scipione del Ferro (1465 – 1526), but he told the solution only to few people, including his student Antonio Maria Fior. Nicolo Fontana Tartaglia, prompted by the rumors, manage to solve the cubic equation x3 +b x2 = -d and made no secret of his discovery. Fior challenged Tartaglia, in 1535, to a public contest, each one had to solve 30 problems proposed by the other in 40 to 50 days. Tartaglia managed to solve his problems of type x3 +m x = n in about two hours, and won the contest. News of Tartaglia victory reached Girolamo Cardan in Milan, where he was preparing to publish Practica Arithmeticae (1539). Cardan invited Tartaglia to visid him and, after much persuasion, made him to divulge his solution of the cubic equation. Tartaglia made Cartan promise to keep the secret until Tartaglia had published it himself.
  • 18. 18 SOLO Complex Variables History of Complex Numbers Girolamo Cardano 1501 - 1576 Nicolo Fontana Tartaglia 1500 - 1557 Solution of cubic equation x3 + b x2 +c x +d = 0 After Tartaglia showed Cardan how to solve cubic equations, Cartan encouraged his student Lodovico Ferrari (1522 – 1565) to use those result and solve quartic equations x4 +p x2 +q x +r=0. Since Tartaglia didn’t publish his results and after hearing from Hannibal Della Nave that Scipione del Ferro first solve cubic equations, Cardan pubished in 1545 in Ars Magna (The Great Art) the solutions of the cubic (credit given to Tartaglia) and quartic equations. This led to another competition between Tartaglia and Cardano, for which the latter did not show up but was represented by his student Lodovico Ferrari. Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and income.
  • 19. 19 SOLO Complex Variables History of Complex Numbers Solution of cubic equation x3 + b x2 +c x +d = 0 023 =+++ dxcxbx 3/bxy −= → 0 27 2 33 393 2 273333 33 2 3 3 22 32 23 23 =−+=      +−+      −+= +−++−+−+−=+      −+      −+      − − nymyb cb dy b cy d cb yc b ybyb b y b ybyd b yc b yb b y nm    Equivalence between x3 + b x2 +c x +d = 0 and y3 +m y = n (depressed cubic equation) Solutions of y3 +m y = n Start from the identity: ( )  ( ) nymybabababa nymy =+→−=−+− 3333 3  0 27273 3 3 36 3 3 333 =−−→−=−==→= m ana a m aban a m bbam 3 32 3,2,13,2,1 2742 mnn wa +±=                       +± −      +±=−= 3 32 3 32 3,2,13,2,1 342 3 3423 mnn m mnn w a m ay 2 31 , 2 31 ,13 3,2,1 ii ew ki k +− === π
  • 20. 20 SOLO Complex Variables History of Complex Numbers Solution of cubic equation x3 + b x2 +c x +d = 0                 ++      +±=                       −      −            + −      +±=                       +± −      +±=−= 3 32 3 32 3,2,1 3 322 3 32 3 32 3,2,1 3 32 3 32 3,2,13,2,1 342342 322 342 3342 342 3 3423 mnnmnn w mnn mnn mmnn w mnn m mnn w a m ay   2 31 , 2 31 ,1 342342 3 3,2,1 3 32 3 32 3,2,13,2,1 ii ew mnnmnn wy ki k +− ==                 +−+      ++= = π Solutions of y3 +m y = n Note: Tartaglia and Cardano knew only the solution w1 = 1
  • 21. 21 SOLO Complex Variables History of Complex Numbers Solution of cubic equation x3 + b x2 +c x +d = 0 A Solution of y3 +m y = n Guess a solution:       −++= 33 huhuwy ( ) uzhuhuhuhuhuhuy 233 3 2333 23 +−=−+      −++−++= Therefore:              +      = = →     −=− = 32 3 2 32 2 3 2 mn h n u mhu nu                 +      −+      +      += 3 32 3 32 3,2,13,2,1 322322 mnnmnn wy Where w are the roots of w3 =1: 2 31 , 2 31 ,13,2,1 ii w +− =
  • 22. 22 SOLO Complex Variables History of Complex Numbers Solution of cubic equation x3 + b x2 +c x +d = 0 Viète Solution of y3 +m y = n François Viète 1540 - 1603 In 1591 François Viète gave another solution to y3 +m y = n Start with identity CC C 34cos3cos43cos 3 cos 3 −=−= = θ θθθ Substitute in y3 +m y = nC m y 3 2 −= 3 3 3 2 1 3 3 3 2 34 3 2 3 8 3       −=−→=−+      −       − m n CCnC m mC m m Assuming we obtain 323 32 1 3 2         ≤      →≤      mn m n φθ πφθ cos 3 2 3cos 23 3 k m n += =      −=               −=      + −=−= − 3 1 3 2 cos 3 2 cos 3 2 3 2 m nkm C m y φ πφ
  • 23. 23 SOLO Complex Variables History of Complex Numbers Solution of cubic equation x3 + b x2 +c x +d = 0 Comparison of Cardano and Viète Solution of y3 +m y = n François Viète 1540 - 1603Cardano solution was               −=      + −=+= − 3 1 3 2 cos 3 2 cos 3 2 m nkm ssy φ πφ Girolamo Cardano 1501 - 15763 32 3 32 342342       +−+      ++= mnnmnn y ( )                              −=       −=       ++= += − + 3 1 2 3 32 3 3 2 cos 3 342 33 m n e m mnn s ssy ki ss φ πφ  or from which we recover Viète Solution 0s φ 2 n 3 3      m 3 s 3 s 0 x 3/φ 1s 2s 0 s 2s 1 s 23 23       −      nm 3 z  120  120  120
  • 24. 24 SOLO Complex Variables History of Complex Numbers Rafael Bombelli 1526 - 1572 John Wallis 1616 - 1703 In 1572 Rafael Bombelli published three of the intended five volumes of “L’Algebra” worked with non-real solutions of the quadratic equation x2 +b x+c=0 by using and where and applying addition and multiplication rules. vu 1−+ vu 1−− ( ) 11 2 −=− In 1673 John Wallis presented a geometric picture of the complex numbers resulting from the equation x2 + b x + c=0, that is close with what we sed today. Wallis's method has the undesirable consequence that is represented by the same point as is 1−− 1− ( )0,b−( )0,b− bb c bb c1 P 2P 2 P 1 P Wallis representation of real roots of quadratics Wallis representation of non-real roots of quadratics
  • 25. 25 Vector Analysis HistorySOLO Caspar Wessel 1745-1818 “On the Analytic Representation of Direction; an Attempt”, 1799 bia + Jean Robert Argand 1768-1822 1806 1−=i 3.R.S. Elliott, “Electromagnetics”,pp.564-568 http://www-groups.dcs.st-and.ac.uk/~history/index.html Wessel's fame as a mathematician rests solely on this paper, which was published in 1799, giving for the first time a geometrical interpretation of complex numbers. Today we call this geometric interpretation the Argand diagram but Wessel's work came first. It was rediscovered by Argandin 1806 and again by Gauss in 1831. (It is worth noting that Gauss redid another part of Wessel's work, for he retriangulated Oldenburg in around 1824.)
  • 26. 26 SOLO Complex Variables History of Complex Numbers Leonhard Euler 1707- 1783 In 1748 Euler published “Introductio in Analysin Infinitorum” in which he introduced the notation and gave the formula1−=i xixeix sincos += In 1751 Euler published his full theory of logarithms and complex numbers. Euler discovered the Cauchy-Riemann equations in 1777 although d’Alembert had discovered them in 1752 while investigating hydrodynamics. Johann Karl Friederich Gauss published the first correct proof of the fundamental theorem of algebra in his doctoral thesis of 1797, but still claimed that "the true metaphysics of the square root of -1 is elusive" as late as 1825. By 1831 Gauss overcame some of his uncertainty about complex numbers and published his work on the geometric representation of complex numbers as points in the plane. Karl Friederich Gauss 1777-1855
  • 27. 27 SOLO Complex Variables History of Complex Numbers Augustin Louis Cauchy )1789-1857( Cauchy is considered the founder of complex analysis after publishing the Cauchy-Riemann equations in 1814 in his paper “Sur les Intégrales Définies”. He created the Residue Theorem and used it to derive a whole host of most interesting series and integral formulas and was the first to define complex numbers as pairs of real numbers. Georg Friedrich Bernhard Riemann 1826 - 1866 In 1851 Riemann give a dissertation in the theory of functions. Return to Table of Contents
  • 28. 28 SOLO Complex Variables Derivatives If f (z) is a single-valued in some region C of the z plane, the derivative of f (z) is defined as: ( ) ( ) ( ) z zfzzf zf z ∆ −∆+ = →∆ 0 lim' provided that the limit exists independent of the manner in which Δ z→0. In such case we say that f (z) is differentiable at z. Analytic Functions If the derivative of f (z) exists at all points of a region C of the z plane, then f (z) is said to be analytic in C. analytic = regular = holomorphic A function f (z) is said to be analytic at a point z0 if there exists a neighborhood |z-z0 | < δ in which f ’ (z) exists. z0 δ Analytic functions have derivatives of any order which themselves are analytic functions. Return to Table of Contents
  • 29. 29 SOLO Complex Variables Derivatives If f (z) is a single-valued in some region C of the z plane, the derivative of f (z) is defined as: ( ) ( ) ( ) z zfzzf zf z ∆ −∆+ = →∆ 0 lim' provided that the limit exists independent of the manner in which Δ z→0. In such case we say that f (z) is differentiable at z. Analytic Functions If the derivative of f (z) exists at all points of a region C of the z plane, then f (z) is said to be analytic in C. analytic = regular = holomorphic A function f (z) is said to be analytic at a point z0 if there exists a neighborhood |z-z0 | < δ in which f ’ (z) exists. z0 δ Analytic functions have derivatives of any order which themselves are analytic functions.
  • 30. 30 SOLO Complex Variables Analytic, Holomorphic, MeromorphicFunctions Return to Table of Contents A Meromorphic Function on an open subset D of the complex plane is a function that is Holomorphic on all D except a set of isolated points, which are poles for the function. (The terminology comes from the Ancient Greek meros (μέρος), meaning “part”, as opposed to holos ( λος)ὅ , meaning “whole”.) The word “Holomorphic" was introduced by two of Cauchy's students, Briot (1817–1882) and Bouquet (1819–1895), and derives from the Greek λοςὅ (holos) meaning "entire", and μορφή (morphē) meaning "form" or "appearance".[2] Today, the term "holomorphic function" is sometimes preferred to "analytic function", as the latter is a more general concept. This is also because an important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow directly from the definitions. The term "analytic" is however also in wide use. The Gamma Function is Meromorphic in the whole complex plane Poles
  • 31. 31 SOLO Complex Variables Cauchy-Riemann Equations A necessary (but not sufficient) condition that f (z) = u (x,y) +i v (x,y) be analytic in a region C, is that u and v satisfy the Cauchy-Riemann equations: y u x v y v x u ∂ ∂ −= ∂ ∂ ∂ ∂ = ∂ ∂ & Proof: Augustin Louis Cauchy )1789-1857( Georg Friedrich Bernhard Riemann 1826 - 1866 ( ) ( ) ( ) z zfzzf zf z ∆ −∆+ = →∆ 0 lim' Provided that the limit exists independent of the manner in which Δ z→0. Choose Δ z = Δ x → ( ) x v i x u zf ∂ ∂ + ∂ ∂ =' Choose Δ z =i Δ y → ( ) y v y u izf ∂ ∂ + ∂ ∂ −=' Equalizing those two expressions we obtain: ( ) y u i y v x v i x u zf ∂ ∂ − ∂ ∂ = ∂ ∂ + ∂ ∂ =' y u x v y v x u ∂ ∂ −= ∂ ∂ ∂ ∂ = ∂ ∂ → & The functions u (x,y) and v (x,y) are called conjugate functions, because if one is given we can find the other (with an arbitrary additive constant). Return to Table of Contents
  • 32. 32 SOLO Complex Variables Harmonic Functions If the second partial derivatives of u (x,y) and v (x,y) with respect to x and y exist and are continuous in a region C, then using Cauchy-Riemann equations, we obtain: 02 2 2 2 2 22 2 2 2 22 = ∂ ∂ + ∂ ∂ →         ∂ ∂ −= ∂∂ ∂ → ∂ ∂ −= ∂ ∂ ∂∂ ∂ = ∂ ∂ → ∂ ∂ = ∂ ∂ ∂∂ ∂ = ∂∂ ∂ ∂ ∂ ∂ ∂ y u x u y u xy v y u x v yx v x u y v x u xyyx y x 02 2 2 2 2 2 2 2 22 22 = ∂ ∂ + ∂ ∂ →         ∂∂ ∂ −= ∂ ∂ → ∂ ∂ −= ∂ ∂ ∂ ∂ = ∂∂ ∂ → ∂ ∂ = ∂ ∂ ∂∂ ∂ = ∂∂ ∂ ∂ ∂ ∂ ∂ y v x v yx u x v y u x v y v xy u y v x u xyyx x y It follows that under those conditions the real and imaginary parts of an analytic function satisfy Laplace’s Equation denoted by: 02 2 2 2 = ∂ Φ∂ + ∂ Φ∂ yx or 2 2 2 2 2 : yxx i xx i x ∂ ∂ + ∂ ∂ =      ∂ ∂ − ∂ ∂ ⋅      ∂ ∂ + ∂ ∂ =∇ where02 =Φ∇ The functions satisfying Laplace’s Equation are called Harmonic Functions. Pierre-Simon Laplace (1749-1827) Return to Table of Contents
  • 33. 33 SOLO Complex Variables Singular Points A point at which f (z) is not analytic is called a singular point. There are various types of singular points: 1. Isolated Singularity The point z0 at which f (z) is not analytic is called an isolated singular point, if we can a neighborhood of z0 in which there are not singular points. z0 δ If no such a neighborhood of z0 can be found then we call z0 a non-isolated singular point. 2. Poles Example ( ) ( ) ( ) ( ) ( ) ( ) ( )5353 1834 32 ++−− ++ = zzzz zz zf has a pole of order 2 at z = 3, a pole of order 3 at z = 5, and two simple poles at z = -3 and z = -5. If we can find a positive integer n such that and is analytic at z=z0 then z = z0 is called a pole of order n. If n = 1, z is called a simple pole. ( ) ( ) 0lim 0 0 ≠=−→ Azfzz n zz ( ) ( ) ( )zfzzz n 0−=ϕ
  • 34. 34 SOLO Complex Variables Singular Points A point at which f (z) is not analytic is called a singular point. There are various types of singular points: 3. Branch Points If f (z) is a multiple valued function at z0, then this is a branch point. Examples: ( ) ( ) n zzzf /1 0−= has a branch point at z=z0 ( ) ( ) ( )[ ]0201ln zzzzzf −−= has a branch points at z=z01 and z=z02 4. Removable Singularities The singular point z0 is a removable singularity of f (z) if exists.( )zf zz 0 lim → Examples: The singular point z = 0 of is a removable singularity z zsin 1 sin lim0 =→ z z z
  • 35. 35 SOLO Complex Variables Singular Points A point at which f (z) is not analytic is called a singular point. There are various types of singular points: 5. Essential Singularities A singularity which is not a pole, branch point or a removable singularity is called an essential singularity. Example: has an essential singularity at z = z0.( ) ( )0/1 zz ezf − = 6. Singularities at Infinity If we say that f (z) has singularities at z →∞. The type of the singularity is the same as that of f (1/w) at w = 0. ( ) 0lim = ∞→ zf z Example: The function f (z) = z5 has a pole of order 5 at z = ∞, since f (1/w) = 1/w5 has a pole of order 5 at w = 0. Return to Table of Contents
  • 36. 36 SOLO Complex Variables Orthogonal Families If f (z) = u (x,y) + i v (x,y) is analytic, then the one-parameter families of curves ( ) ( ) βα == yxvyxu ,,, where α and β are constant are orthogonal. Proof: The normal to u (x,y) = α is: ( ) y y u x x u yxu 11, ∂ ∂ + ∂ ∂ =∇ The normal to v (x,y) = β is: ( ) y y v x x v yxv 11, ∂ ∂ + ∂ ∂ =∇ The scalar product between the normal to u (x,y) = α and the normal to v (x,y) = β is: ( ) ( ) y v y u x v x u yxvyxu ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ =∇⋅∇ ,, Using the Cauchy-Riemann Equation for the analytic f (z): y u x v y v x u ∂ ∂ −= ∂ ∂ ∂ ∂ = ∂ ∂ & ( ) ( ) 0,, = ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ −=∇⋅∇ y v y u y u y v yxvyxu x y ( ) α=yxu , ( ) β=yxv , planez u v planew Return to Table of Contents
  • 37. 37 SOLO Complex Variables Complex Line Integrals Let f (z) be continuous at all points on a curve C of a finite length L. ( ) ( ) ( )∑∑ == − ∆=−= n i ii n i iiin zfzzfS 11 1 ξξ C 1 z nzb = 2z 0 za = 1−iz iz 1 ξ 2 ξ i ξ n ξ Let subdivide C into n parts by n arbitrary points z1, z2,…,zn, and call a=z0 and b=zn. On each arc joining zi-1 to zi choose a point ξi. Define the sum: Let the number of subdivisions n increase in such a way that the largest of Δzi approaches zero, then the sum approaches a limit that is called the line integral (also Riemann-Stieltjes integral). ( ) ( ) ( )∫∫∑ ==∆= = →∆∞→ C b a n i ii z nn zdzfzdzfzfS i 1 0 limlim ξ Properties of Integrals ( ) ( )[ ] ( ) ( )∫∫∫ +=+ CCC zdzgzdzfzdzgzf ( ) ( ) constantAzdzfAzdzfA CC == ∫∫ ( ) ( )∫∫ −= a b b a zdzfzdzf ( ) ( ) ( )∫∫∫ += b c c a b a zdzfzdzfzdzf ( ) ( ) ( ) CoflengthLandConMzfLMzdzfzdzf CC ≤≤≤ ∫∫ Return to Table of Contents
  • 38. 38 SOLO Complex Variables Simply and Multiply Connected Regions A region R is called simply-connected if any simple closed curve Γ, which lies in R can be shrunk to a point without leaving R. A region R that is not simply-connected is called multiply-connected. C0 x y R C1 Γ C0 x y R C1 C2 C3 Γ C x y R Γ C x y R Γsimply-connected multiply-connected. Return to Table of Contents
  • 39. 39 SOLO Complex Variables Green’s Theorem in the Plane C R Let P (x,y) and Q (x,y) be continuous and have continuous partial derivatives in a region R and on the boundary C. Green’s Theorem states that: GEORGE STOCKES 1819-1903 A more general theorem was given by Stokes ( ) ∫∫∫       ∂ ∂ − ∂ ∂ =+ R dydx y P x Q dyQdxP C ( ) ∫∫∫∫∫∫∫       ∂ ∂ − ∂ ∂ +      ∂ ∂ − ∂ ∂ +      ∂ ∂ − ∂ ∂ =++ yzxzxy RRR dzdy z Q y R dzdx x R z P dydx y P x Q dzRdyQdxP C or in vector form: ∫∫∫ ⋅×∇=⋅ S dAFdrF C  where: ( ) ( ) ( ) ( ) zzyxRyzyxQxzyxPzyxF 1,,1,,1,,,, ++=  zdzydyxdxdr 111 ++= zdydxydzdxxdzdydA 111 ++= GEORGE GREEN 1793-1841 z z y y x x 111 ∂ ∂ + ∂ ∂ + ∂ ∂ =∇
  • 40. 40 SOLO Complex Variables Proof of Green’s Theorem in the Plane C R P T S Q a b x y ( )xgy 2= ( )xgy 1= Start with a region R and the boundary curve C, defined by S,Q,P,T, where QP and TS are parallel with y axis. ( ) ( ) ∫ ∫∫∫ = = ∂ ∂ = ∂ ∂ b a xgy Xgy dy y P dxdydx y P 2 R By the fundamental lemma of integral calculus: ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )[ ]xgxPxgxPyxPdy y yxP xgy xgy xgy Xgy 12 ,,, , 2 1 2 −== ∂ ∂ = = = = ∫ Therefore: ( )[ ] ( )[ ]∫∫∫∫ −= ∂ ∂ b a b a dxxgxPdxxgxPdydx y P 12 ,, R but: ( )[ ] ( )[ ]∫∫ = a bSQ dxxgxPdxxgxP 22 ,, integral along curve SQ ( )[ ] ( )[ ]∫∫ = b aPT dxxgxPdxxgxP 11 ,, integral along curve PT If we add to those integrals: ( ) ( ) 00,, === ∫∫ dxsincedxyxPdxyxP QPTS we obtain: ( )[ ] ( ) ( )[ ] ( ) ( )∫∫∫∫∫∫∫ −=−−−−= ∂ ∂ CTSPTQPSQ dxyxPdxyxPdxxgxPdxyxPdxxgxPdydx y P ,,,,, 12 R Assume that PT is defined by the function y = g1 (x) and SQ is defined by the function y = g2 (x), both smooth and y P ∂ ∂ is continuous in R:
  • 41. 41 SOLO Complex Variables Proof of Green’s Theorem in the Plane (continue – 1) In the same way: Therefore we obtain: ( )∫∫∫ −= ∂ ∂ C dxyxPdydx y P , R ( )∫∫∫ = ∂ ∂ C dyyxQdydx x Q , R ( ) ∫∫∫       ∂ ∂ − ∂ ∂ =+ R dydx y P x Q dyQdxP C The line integral is evaluated by traveling C counterclockwise. For a general single connected region, as that described in Figure to the right, can be divided in a finite number of sub-regions Ri, each of each are of the type described in the Figure above. Since the adjacent regions boundaries are traveled in opposite directions, there sum is zero, and we obtain again: ( ) ∫∫∫       ∂ ∂ − ∂ ∂ =+ R dydx y P x Q dyQdxP C C R4 x y R R3 R1 R2 C R P T S Q a b x y ( )xgy 2= ( )xgy 1=
  • 42. 42 SOLO Complex Variables Proof of Green’s Theorem in the Plane (continue – 2) The general multiply-connected regions can be transformed in a simply connected region by infinitesimal slits Since the slits boundaries are traveled in opposite directions, there integral sum is zero: C0 x y R C1 P0 P1 C0 x y R C1 C2 C3 ( ) ( ) ∫∫∑ ∫∫       ∂ ∂ − ∂ ∂ =+−+ R dydx y P x Q dyQdxPdyQdxP i CC i0 All line integrals are evaluated by traveling Ci i=0,1,… counterclockwise. ( ) ( ) 0 0 1 1 0 =+++ ∫∫ P P P P dyQdxPdyQdxP We obtain: Return to Table of Contents
  • 43. 43 SOLO Complex Variables Consequences of Green’s Theorem in the Plane Let P (x,y) and Q (x,y) be continuous and have continuous first partial derivative at each point of a simply-connected region R. A necessary and sufficient condition that around every closed path C in R is that in R. This is synonym to the condition that is path independent. y P x Q ∂ ∂ = ∂ ∂ ( ) 0=+∫C dyQdxP Sufficiency: Suppose y P x Q ∂ ∂ = ∂ ∂ According to Green’s Theorem ( ) 0=      ∂ ∂ − ∂ ∂ =+ ∫∫∫ R dydx y P x Q dyQdxP C Necessity: 0<> ∂ ∂ − ∂ ∂ or y P x Q Suppose along every path C in R. Assume that at some point (x0,y0) in R. Since ∂ Q/ ∂ x and ∂ P/ ∂ y are continuous exists a region τ around (x0,y0) and boundary Γ for which , therefore: ( ) 0=+∫C dyQdxP 0<> ∂ ∂ − ∂ ∂ or y P x Q ( ) 0<>      ∂ ∂ − ∂ ∂ =+ ∫∫∫Γ ordydx y P x Q dyQdxP τ C x y R ( )∫ + L dyQdxP 0= ∂ ∂ − ∂ ∂ y P x Q This is a contradiction to the assumption, therefore q.e.d. Return to Table of Contents
  • 44. 44 SOLO Complex Variables Cauchy’s Theorem C x y R Proof: ( ) 0=∫C dzzf If f (z) is analytic with derivative f ‘ (z) which is continuous at all points inside and on a simple closed curve C, then: ( ) ( ) ( )yxviyxuzf ,, +=Since is analytic and has continuous first order derivative ( ) y u i y v x v i x u zd fd zf iyzxz ∂ ∂ − ∂ ∂ = ∂ ∂ + ∂ ∂ == == ' y u x v y v x u ∂ ∂ −= ∂ ∂ ∂ ∂ = ∂ ∂ & Cauchy - Riemann ( ) ( ) ( ) ( ) ( ) 0 00 =      ∂ ∂ − ∂ ∂ +      ∂ ∂ − ∂ ∂ −= ++−=++= ∫∫∫∫ ∫∫∫∫ RR dydx y v x u idydx y u x v dyudxvidyvdxudyidxviudzzf CCCC    q.e.d. Augustin Louis Cauchy )1789-1857( Return to Table of Contents
  • 45. 45 SOLO Complex Variables Cauchy-Goursat Theorem C x y R Proof: ( ) 0=∫C dzzf If f (z) is analytic which is continuous at all points inside and on a simple closed curve C, then: Augustin Louis Cauchy )1789-1857( Goursat removed the Cauchy’s condition that f ‘ (z) should be continuous in R. C F DE A B I ∆ IV ∆ II ∆ III ∆ Start with a triangle ABC in z in which f (z) is analytic, Join the midpoints E,D,F to obtain four equal triangles ΔI, ΔII, ΔIII, ΔIV. We have: ( ) ( ) ( ) ( ) ( )∫∫∫∫ ∫∫∫∫ ∫∫∫∫∫∫∫∫∫ ∫∫∫∫ ∆∆∆∆ +++= +++=         +++         ++         ++         += ++= IVIIIIII dzzfdzzfdzzfdzzf dzzf DEFDFCDFEBFEDAED FDEFDEDFFCDFEEBFEDDAE FCDEBFDAEABCA Eduard Jean-Baptiste Goursart 1858 - 1936
  • 46. 46 SOLO Complex Variables Proof of Cauchy-Goursat Theorem (continue – 1) If f (z) is analytic which is continuous at all points inside and on a simple closed curve C, then: C F DE A B I ∆ IV∆ II ∆ III ∆ ( ) ( ) ( ) ( ) ( )∫∫∫∫∫ ∆∆∆∆ +++= IVIIIIII dzzfdzzfdzzfdzzfdzzf ABCA then: ( ) ( ) ( ) ( ) ( )∫∫∫∫∫ ∆∆∆∆ +++≤ IVIIIIII dzzfdzzfdzzfdzzfdzzf ABCA Let Δ1 be the triangle in which the absolute value of the integral is maximum. ( ) ( )∫∫ ∆∆ ≤ 1 4 dzzfdzzf Continue this procedure in triangle Δ1 in which Δ2 is the triangle in which the absolute value of the integral is maximum. ( ) ( ) ( )∫∫∫ ∆∆∆ ≤≤ 21 2 44 dzzfdzzfdzzf ( ) ( )∫∫ ∆∆ ≤ n dzzfdzzf n 4
  • 47. 47 SOLO Complex Variables Proof of Cauchy-Goursat Theorem (continue – 2) C F DE A B I ∆ IV ∆ II ∆ III ∆ ( ) ( )∫∫ ∆∆ ≤ n dzzfdzzf n 4 For an analytic function f (z) compute ( ) ( ) ( ) ( )0 0 0 0 ':, zf zz zfzf zz − − − =η ( ) ( ) ( ) ( ) ( ) ( ) 0'''lim,lim 000 0 0 0 00 =−=       − − − = →→ zfzfzf zz zfzf zz zzzz η ( ) ( ) ( ) ( ) ( ) ( ) ( )0000000 ,'&,..,0 zzzzzzzfzfzfzzwheneverzzts −+−+=<−<∃>∀ ηδεηδε ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )∫∫∫∫ ∆∆ ← ∆∆ −≤−+−+≤ nnnn dzzzzzdzzzzzdzzzzfzfdzzf TheoremIntegralCauchy 0000 0 )00 ,,' ηη    n∆ 0z na nb nc z 0 zz − 0 zzcbaP nnnn −≥++= ( ) ( ) ( ) 2 2 00 2 ,       ==≤−≤ ∫∫∫ ∆∆∆ nnn P PdzPdzzzzzdzzf nnn εεεη But , where Pn the perimeter of Δn and P the perimeter of Δ are related, by construction, by ( ) δεη <≤−< n Pzzzz 00 &, n n PP 2/= q.e.d. ( ) ( ) ( ) 0 4 44 0 2 2 =→=≤≤ ∫∫∫ ∆ → ∆∆ dzzfP P dzzfdzzf n nn n ε εε
  • 48. 48 SOLO Complex Variables Proof of Cauchy-Goursat Theorem (continue – 3) n z1 z 2 z 1−i z i z 1−n z n ∆1∆ 2∆ 3 ∆ i ∆ C O q.e.d. For the general case of a simple closed curve C we take n points on C: z1, z2,…,zn and a point O inside C. We obtain n triangles Δ1, Δ2,.., Δn, for each of them we proved Cauchy-Goursat Theorem. Let define the sum: ( ) ∑= − − ∆= n i zz iin ii zzfS 1 1 : we have: ( ) ( ) ( ) ( ) 0 1 1 =++= ∫∫∫∫ − −∆ i i i ii z O O z z z dzzfdzzfdzzfdzzf ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( )    n i i i i i i i ii S n i z z i n i z z i n i z z ii n i z z n i dzzfdzzfzfdzzfzfzfdzzfdzzf ∑ ∫∑∫∑ ∫∑ ∫∑ ∫ ===== ∆ −−−− +−=+−=== 11111 1111 0 ( ) ( ) ( )ε ε NnforSdzzfdzzfS n CC nn ><−→= ∫∫∞→ 2 lim ( ) ( )[ ] ( ) ( )[ ] ( ) 221 1 111 11 εε =−≤−≤−≤→−= ∑∑∑ ∫∑∫ = − === −− n i ii n i i n i z z in n i z z in zz L dzfzfdzzfzfSdzzfzfS i i i i ( ) ( ) ( ) ( ) 0 22 =→>=+<+−≤ ∫∫∫ C nn CC dzzfNnforSSdzzfdzzf εε εε Since we proved that , we can write:( ) 0=∫∆ dzzf Return to Table of Contents
  • 49. 49 SOLO Complex Variables Consequences of Cauchy-Goursat Theorem B x y R A C1 C2 D1 D2 a bIf f (z) is analytic in a simply-connected region R, then is independent of the path in R joining any two points a and b in R. ( )∫ b a dzzf Let look at thr closed path AC1D1BD2C2A in R inside which f (z) is analytic. According to Cauchy-Goursat Theorem ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫∫∫∫∫ =→−=+== BDAcBDAcBDAcBDAcACBDBDAcACBDDAC dzzfdzzfdzzfdzzfdzzfdzzfdzzf 2211221122111122 0 Proof: If f (z) is analytic in a multiply-connected region R, bounded by two simple closed curves C1 and C2, then: 1 2 C1 x y R C2 P0 P1 ( ) ( )∫∫ = 21 CC dzzfdzzf The general multiply connected regions can be transformed in a single connected region by an infinitesimal slit P0 to P1. ( ) ( ) ( ) ( ) ( ) ( )∫∫∫∫∫∫ =→+++= 212 0 1 1 01 0 0 CCC P P P PC dzzfdzzfdzzfdzzfdzzfdzzf    Proof: Return to Table of Contents
  • 50. 50 SOLO Complex Variables Cauchy’s Integral Formulas Augustin Louis Cauchy )1789-1857( If f (z) is analytic inside and on a simple closed curve C and a is any point inside C then ( ) ( ) ∫ − = C dz az zf i af π2 1 C x y R a Γ Proof: Let chose a circle Γ with center at a [ ]{ }πθε θ 2,0,: ∈+==Γ i eazz Since f (z)/ (z-a) is analytic in the region defined between C and the circle Γ we can use: ( ) ( ) ∫∫ Γ − = − zd az zf zd az zf C ( ) ( ) ( ) ( )afidafidei e eaf zd az zf i i i πθθε ε ε ππ θ θ θ ε 21lim 2 0 2 0 0 === + = − ∫∫∫ → Γ therefore: ( ) ( ) ∫ − = C dz az zf i af π2 1 q.e.d. Cauchy’s Integral Formulas and Related Theorems Return to Table of Contents
  • 51. 51 SOLO Complex Variables Cauchy’s Integral Formulas for the n Derivative of a Function Augustin Louis Cauchy )1789-1857( If f (z) is analytic inside and on a simple closed curve C and a is any point inside C, where the n derivative exists, then ( ) ( ) ( ) ( )∫ + − = C n n dz az zf i n af 1 2 ! π C x y R a Γ Proof: Let prove this by induction. Assume that this is true for n-1: Then we can differentiate under the sign of integration: ( ) ( ) ∫ − = C dz az zf i af π2 1 For n = 0 we found ( ) ( ) ( ) ( ) ( )∫ − − =− C n n dz az zf i n af π2 !11 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ + − − − = − − == C n C n nn dz az n zf i n dz azad d zf i n af ad d af 1 1 2 !11 2 !1 ππ q.e.d.Therefore for n we obtain: ( ) ( ) ( ) ( )∫ + − = C n n dz az zf i n af 1 2 ! π We can see that an analytic function has derivatives of all orders. Return to Table of Contents
  • 52. 52 SOLO Complex Variables Morera’s Theorem (the converse of Cauchy’s theorem) If f (z) is continuous in a simply-connected region R and if around every simple closed curve C in R then f (z) ia analytic in R. ( ) 0=∫C dzzf B x y R A C1 C2 D1 D2 a z Proof: Since around every closed curve C in R( ) 0=∫C dzzf ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫∫∫∫∫ =→−=+== BDAcBDAcBDAcBDAcACBDBDAcACBDDAC dzzfdzzfdzzfdzzfdzzfdzzfdzzf 2211221122111122 0 The integral is independent on path between two points, if the path is in R ( ) ( )∫= z a dzzfzF Let choose a straight path between z and z+Δz ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∫∫∫ ∆+∆+ − ∆ =−         − ∆ =− ∆ −∆+ zz z z a zz a udzfuf z zfudufuduf z zf z zFzzF 11 Since f (z) is continuous ( ) ( ) δε <−≤− zuwheneverzfuf Therefore ( ) ( ) ( ) ( ) ( ) ( ) ( )zf zd zFd zudzfuf z zf z zFzzF zz z =→<∆∀=− ∆ ≤− ∆ −∆+ ∫ ∆+ δε 1 C x y R z Νδz+∆ z Since F (z) has a derivative in R, it is analytic, and so are its derivatives, i.e. f (z) Return to Table of Contents Giacinto Morera 1856 - 1907
  • 53. 53 SOLO Complex Variables Cauchy’s Inequality Augustin Louis Cauchy )1789-1857( If f (z) is analytic inside and on a circle C of radius r and center at z = a, then ( ) ( ) ,...2,1,0 ! = ⋅ ≤ n r nM af n n where M is a constant such that | f (z) |< M is an upper bound of | f (z) | on C. Proof: C x y a r Use Cauchy Integral Formula: ( ) ( ) ( ) ( ) ,2,1,0 ! 2 1 2 !1 2 ! 2 ! 1 2 0 11 ===≤ − ≤ ++ < + ∫∫ n r nM r r Mn dr r Mn dz az zfn af nnn Mzf C n n π π θ ππ π ( ) ( ) ( ) ( ) ,2,1,0 2 ! 1 = − = ∫ + ndz az zf i n af C n n π q.e.d. On the circle C: .θi eraz += Return to Table of Contents
  • 54. 54 SOLO Complex Variables Liouville’s Theorem Joseph Liouville 1809 - 1882 If for all z in the complex plane: (1) f (z) is analytic (2) f (z) is bounded, i.e. | f (z) |< M for some constant M then f (z) must be a constant. Proof No. 1: Using Cauchy’s Inequality: ( ) ( ) ,...2,1,0 ! = ⋅ ≤ n r nM af n n Letting n=1 we obtain: ( ) r M af ≤' Since f (z) is analytic in all z plane we can take r → ∞ to obtain ( ) ( ) ( ) constantafaaf r M af r =→∀=→=≤ ∞→ ,00lim '' q.e.d.
  • 55. 55 SOLO Complex Variables Liouville’s Theorem Joseph Liouville 1809 - 1882 If for all z in the complex plane: (1) f (z) is analytic (2) f (z) is bounded, i.e. | f (z) |< M for some constant M then f (z) must be a constant. Proof No. 2: Using Cauchy’s Integral Formula q.e.d. Let a and b be any two points in z plane. Draw a circle C with center at a and radius r > 2 | a-b | C x y a r b 2/rba <− ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ −− − = − − − =− CCC dz bzaz zf i ab dz az zf i dz bz zf i afbf πππ 22 1 2 1 We have raz =− ( ) ( ) 2/rbarbaazbaazbz ≤−−=−−−≥−+−=− ( ) ( ) ( ) ( ) ( ) ( ) ( ) r Mab dr rr Mab dz bzaz zfab dz bzaz zfab afbf CC − = − ≤ −− − ≤ −− − =− ∫∫∫ 2 2/222 2 0 π θ πππ Since f (z) is analytic in all z plane we can take r → ∞ to obtain ( ) ( ) ( ) ( )afbfafbf =→=− 0 therefore f (z) is constant. Return to Table of Contents
  • 56. 56 SOLO Complex Variables Foundamental Theorem of Algebra Every polynomial equation P (z) = a0 + a1z+a2z2 +…+anzn =0 with degree n ≥ 1 and an ≠ 0 (ai are complex constants) has at least one root. From this it follows that P (z) = 0 has exactly n roots, due attention being paid to multiplicities of roots. Proof: If P (z) = 0 has no root, then f (z) = 1 / P (z) is analytic for all z. Also | f (z) |= 1 / | P (z) | is bounded. Then by Liouville’s Theorem f (z) and then P (z) are constant. This is a contradiction to the fact that P (z) is a polynomial in z, therefore P (z) = 0 must have at least one root (zero). Suppose that z = a is one root of P (z) = 0. Hence P (a) = 0 and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )zQazazaazaaza aaaaaaazazazaaaPzP nn n n n n n −=−++−+−= ++++−++++=−   22 21 2 210 2 210 Since an ≠ 0, Q (s) is a polynomial of degree n-1. Applying the same reasoning to the polynomial Q (s) of degree n-1, we conclude that it must also have at least one root. This procedure continues until n = 0, therefore it follows that P (z) has exactly n roots. q.e.d. Return to Table of Contents
  • 57. 57 SOLO Complex Variables Gauss’ Mean Value Theorem Karl Friederich Gauss 1777-1855 C x y a r If f (z) is analytic inside and on a circle C with center at a and radius r, then f (a) is the mean of the values of f (z) on C, i.e., ( ) ( ) . 2 1 2 9 ∫ += π θ θ π derafaf i Proof: Use Cauchy Integral Formula: On the circle C: θθ ii eridzeraz =+= . ( ) ( ) ∫ − = C dz az zf i af π2 1 ( ) ( ) ( ) ( )∫∫∫ += + = − = θ π θ ππ θθ θ θ derafderi er eraf i dz az zf i af ii i i C 2 1 2 1 2 1 q.e.d. Return to Table of Contents
  • 58. 58 SOLO Complex Variables Maximum Modulus Theorem If f (z) is analytic inside and on a simple closed curve C and is not identically equal to a constant, then the maximum value of | f (z) | occurs on C. Proof: The proof is based on the continuity of f (z) and on the Gauss’ Mean Value Theorem. C x y a r C C1 C2 C3 R α b Since f (z) is analytic in C, | f (z) | has a maximum M inside or on C. Suppose that the maximum value is achieved at the point a inside C, i.e. | f (a) |=M =max | f (z) |. Since a is inside C we can find a circle C1, with center at a that is inside C. Since f (z) is not constant we can find a point b in C1 such that | f (b)|=M – ε< | f (a)|. Using the continuity of f (z) we can find a circle around b, C2, ( ) ( ){ }δε <−<−= bzforbfzfzC 2/:2 ( ) ( ) 2/2/2/ εεεε −=+−=+< MMbfzf Now apply the Gauss’ Mean Value Theorem for point a and the circle with center at a passing trough b, C3. Define by α the arc of C3 inside C2.( ) ( ) . 2 1 2 9 ∫ += π θ θ π derafaf i ( ) ( ) ( ) ( ) ( ) ( ) π εα απ ππ α εθθθ π π α θ α ε θ π θ 4 2 22 2/. 2 1 2 9 2/ 2 9 −=−+−≤+++≤+== ∫∫∫ ≤−≤ M M MderafderafderafMaf M i M ii 
  • 59. 59 SOLO Complex Variables Maximum Modulus Theorem If f (z) is analytic inside and on a simple closed curve C and is not identically equal to a constant, then the maximum value of | f (z) | occurs on C. Proof (continue): ( ) ( ) ( ) ( ) ( ) ( ) π εα απ ππ α εθθθ π π α θ α ε θ π θ 4 2 22 2/. 2 1 2 9 2/ 2 9 −=−+−≤+++≤+== ∫∫∫ ≤−≤ M M MderafderafderafMaf M i M ii  We obtained that is impossible, therefore a for which |f (z)| is maximum cannot be inside C, but on C. ( ) π εα 4 ? −== MMaf C x y a r C C1 C2 C3 R α b q.e.d. Return to Table of Contents
  • 60. 60 SOLO Complex Variables Minimum Modulus Theorem If f (z) is analytic inside and on a simple closed curve C and f (z) ≠ 0 inside C then | f (z) | assumes its minimum value on C. Proof: Since f (z) is analytic inside and on a simple closed curve C and f (z) ≠ 0 inside C it follows that 1/ f (z) is analytic inside and on C. Then according to Maximum Modulus Theorem 1/| f (z) | assumes its maximum vale on C and therefore | f (z) | assumes its minimum value on C. q.e.d. x y C R Return to Table of Contents
  • 61. 61 SOLO Complex Variables Poisson’s Integral Formulas for a Circle Siméon Denis Poisson 1781-1840 Let f (z) be analytic inside and on the circle C defined by |z| = R, and let z = r e iθ be any point inside C, then: ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ − − = − − = +−− − == π π φ φ φ θφ π φθ φ π φ π φ φθπ 2 0 2 0 2 22 2 22 2 0 22 22 2 1 2 1 cos22 1 deRf zeR zR deRf ereR rR deRf rrRR rR erzf i i i ii ii C x y R R z' rR2/r θ ∗ z z r θ− ∗ = zRz /2 1 Proof: Since f (z) is analytic in C we can apply Cauchy’s Integral Formula: ( ) ( ) ( ) ∫ − == C i dz zz zf i erfzf ' ' ' 2 1 π θ ( ) ∫ ∗ − = C dz zRz zf i ' /' ' 2 1 0 2 π If we subtract those equations we obtain: The inverse of the point z with respect to C is and lies outside C, therefore by Cauchy’s Theorem: ∗ = zRz /2 1 ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∗∗ −− − =      − − − == CC i dzzf zRzzz zRz i dzzf zRzzzi erfzf '' /'' / 2 1 '' /' 1 ' 1 2 1 2 2 2 ππ θ
  • 62. 62 SOLO Complex Variables Poisson’s Integral Formulas for a Circle Siméon Denis Poisson 1781-1840 Proof (continue): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ ∫ +−− − = −− − = −− − = −− − = −− − == −− + ∗ π φ π φ θφθφ π φ θφθφ φθ π φφ θφθφ θθ θ φ φθπ φ π φ π φ π π 2 0 22 22 2 0 22 2 0 22 2 0 2 2 2 2 cos22 1 2 1 2 1 / / 2 1 '' /'' / 2 1 deRf RrRr rR deRf ereRereR rR deRf eRerereR eRr deRieRf erReRereR erRer i dzzf zRzzz zRz i erfzf i i iiii i iiii i ii iiii ii C i Writing we have:( ) ( ) ( )θθθ ,, rviruerf i += ( ) ( ) ( ) ( )∫ +−− − = π φφ φθπ θ 2 0 22 22 , cos22 1 , dRu RrRr rR ru ( ) ( ) ( ) ( )∫ +−− − = π φφ φθπ θ 2 0 22 22 , cos22 1 , dRv RrRr rR rv q.e.d. C x y R R z' rR2/r θ ∗ z z r θ− ∗ = zRz /2 1 Return to Table of Contents
  • 63. 63 SOLO Complex Variables Poisson’s Integral Formulas for a Half Plane Siméon Denis Poisson 1781-1840 C x y R ∗ z z R Let f (z) be analytic in the upper half y ≥ 0 of the z plane and let z = (x + i y) any point in this upper half plane, then: ( ) ( ) ( )∫ +∞ ∞− +− = dw yxw wfy zf 22 Proof: Let C be the boundary of a a semicircle of radius R containing as an interior point, but does not contain yixz += yixz −=∗ Using Cauchy’s Integral Formula we have: ( ) ( ) ∫ − = C dw zw wf i zf π2 1 By subtraction we obtain: ( ) ∫ ∗ − = C dw zw wf iπ2 1 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ +− = +−−− −−+ = −− − =      − − − = ∗∗ CC CC dwwf yxw yi i dwwf yixwyixw yixyix i dwwf zwzw zz i dwwf zwzwi zf 22 2 2 1 2 1 2 111 2 1 ππ ππ
  • 64. 64 SOLO Complex Variables Poisson’s Integral Formulas for a Half Plane Siméon Denis Poisson 1781-1840 Proof (continue): Where Γ is the upper a semicircle of radius R. ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫ Γ + − +− + +− = +− = dwwf yxw y dwwf yxw y dwwf yxw yi i zf R R C 2222 22 11 2 2 1 ππ π If we take R→∞ we obtain: ( ) ( ) 0lim 1 22 = +−∫Γ ∞→ dwwf yxw y R π ( ) ( ) ( )∫ +∞ ∞− +− = dw yxw wfy zf 22 Therefore: ( ) ( ) ( )∫ +∞ ∞− +− = dw yxw wufy yxu 22 0, , ( ) ( ) ( )∫ +∞ ∞− +− = dw yxw wvfy yxv 22 0, , Writing and since w varies on x axis , and we have: ( ) ( ) ( )yxviyxuzf ,, += ( ) ( ) ( )0,0, wviwuwf += q.e.d. C x y R ∗ z z R Return to Table of Contents
  • 65. 65 SOLO Infinite Series Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1Convergence Definition: The series Sn converges to S as n →∞ if for all ε > 0 there exists an positive integer N such that If no such N exists then we say that the series diverges. NnallforSuSS n i in ><−=− ∑= ε 1 Convergence Theorem: The series Sn converges as n →∞ if and only if there exists an positive integer M such that If no such M exists then we say that the series diverges. 1 1 ><= ∑= NallforMuS N i iN If S is unknown we can use the Cauchy Criterion for convergence: for all ε > 0 there exists an positive integer N such that NmnallforuuSS m j j n i imn ><−=− ∑∑ == , 11 ε Augustin Louis Cauchy )1789-1857( A necessary (but not sufficient) condition for convergence is that lim i→∞ ui = 0 Return to the Table of Content Cauchy Convergence Criterion
  • 66. 66 SOLO Infinite Series Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Convergence Tests In term by term a series of terms 0 ≤ un ≤ an, in which the an form a convergent series, then is also convergent.∑n nu Return to the Table of Content
  • 67. 67 SOLO The Geometric Series ( ) ( ) ( ) ( ) ( ) ( ) r r a r rrarararaa r rS nn nG − − = − −⋅+++++ = − − − − 1 1 1 1 1 1 132 1  Multiply and divide by (1 – r) ( ) ∑ − = − − =+++++= 1 0 132 1 n i in nG rararararaaS  We can see that ( )      ≥∞ < −= − − = ∞→ − ∞→ diverger converger r a r r aS n n nG n 1 1 1 1 1 limlim 1 Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series
  • 68. 68 SOLO Convergence Tests Cauchy Root Test Augustin Louis Cauchy )1789-1857( If (an) 1/n ≤ r < 1 for all sufficiently large n, with r independent of n, then is convergent. If (an) 1/n ≥ 1 for all sufficiently large n, then is divergent. ∑n na ∑n na The first part of this test is verified easily by raising (an) 1/n ≤ r to the nth power. We get: 1<≤ n n ra ∑n naSince rn is just the nth term in a Convergent Geometric Series, is convergent by the Comparison Test. Conversely, if (an) 1/n ≥ 1, the an ≥ 1 and the series diverge. This Root Test is particularly useful in establishing the properties of Power Series. Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series Return to the Table of Content
  • 69. 69 SOLO Convergence Tests D’Alembert or Cauchy Ratio Test Jean Le Rond D’Alembert 1717 - 1783 If (an+1/an) ≤ r < 1 for all sufficiently large n, with r independent of n, then is convergent. If (an+1/an) ≥ 1 for all sufficiently large n, then is divergent. ∑n na ∑n na Convergence is proved by direct comparison with the geometric series (1+r+r2 + … )      = > < + ∞→ ateindetermin,1 ,1 ,1 lim 1 divergence econvergenc a a n n n ∑= = n i n n nS 1 2/Example: convergent n n a a n nn n n n 2 12 2 1 limlim 1 1 = + = +∞→ + ∞→ Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series Return to the Table of Content Augustin Louis Cauchy )1789-1857(
  • 70. 70 SOLO Convergence Tests Maclaurin or Euler Integral Test Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series Return to the Table of Content Let f (n) = an, i.e. f (x) is a monotonic decreasing function. Then converges if is finite and diverges if the integral is infinite. ∑n na ( )∫ ∞ 1 xdxf ( ) ( ) ( )1 111 fxdxfaxdxf n n +≤≤ ∫∑∫ ∞∞ = ∞ Colin Maclaurin 1698 - 1746 Leonhard Euler (1707 – 1`783) Is geometrically obvious that: ( )xf x 1 2 3 4 ( ) 11 af = ( ) 22 af = Comparison of Integral and Sum-Blocks Leading ( )xf x1 2 3 4 ( ) 11 af = Comparison of Integral and Sum-Blocks Lagging
  • 71. SOLO Convergence Tests Kummer’s Test Consider a Series of positive terms ui and a sequence of positive constants ai. If for all n ≥ N, where N is some fixed number, then converges. If and diverges, then diverges. The two tests can be written as: Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series ∑ ∞ =1i iu 01 1 >≥− + + Ca u u a n n n n 01 1 ≤− + + n n n n a u u a ∑ ∞ =1i iu∑ ∞ = − 1 1 i ia    ∞→< > =      − −+ + ∞→ divergea converge Ca u u a i n n n n n 11 1 &0 0 lim Ernst Eduard Kummer (1810 – 1893)
  • 72. 72 SOLO Convergence Tests Kummer’s Test (continue – 1) Consider a Series of positive terms ui and a sequence of positive constants ai. If for all n ≥ N, where N is some fixed number, then converges. 01 1 >≥− + + Ca u u a n n n n ∑ ∞ =1i iu Ernst Eduard Kummer (1810 – 1893) nnnnn NNNNN NNNNN uauauC uauauC uauauC −≤ −≤ −≤ −− +++++ +++ 11 22112 111  Proof: Add and divide by C C ua C ua u nnNNn Ni i −≤∑ += 1 C ua u C ua C ua uuuuS NNN i i nnNNN i i n Ni i N i i n i in +<−+≤+== ∑∑∑∑∑ ==+=== 11111 The partial sums Sn have an upper bound. Since the lower bound is zero the sum must converge.∑ iu q.e.d. Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series
  • 73. 73 SOLO Convergence Tests Kummer’s Test (continue – 2) Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series Consider a Series of positive terms ui and a sequence of positive constants ai. If and diverges, then diverges. 01 1 ≤− + + n n n n a u u a ∑ ∞ =1i iu∑ ∞ = − 1 1 i ia Proof: Nnuauaua NNnnnn >≥≥≥ −− ,11  Since an > 0 n NN n a ua u ≥ and ∑∑ ∞ += − ∞ += ≥ 1 1 1 Ni iNN Ni i auau If diverges, then by comparison test diverges.∑ ∞ = − 1 1 i ia ∑ ∞ =1i iu Return to the Table of Content Ernst Eduard Kummer (1810 – 1893)
  • 74. 74 SOLO Convergence Tests Raabe’s Test If un > 0 and if for all n ≥ N, where N is a positive integer independent on, then converges. If Then diverges (as diverges). The limit form of Raabe’s test is Proof: In Kummer’s Test choose an = n and P = C + 1. Joseph Ludwig Raabe (1801 – 1859) Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series Return to the Table of Content 11 1 >≥      − + P u u n n n ∑i iu 11 1 ≤      − +n n u u n ∑i iu ∑ − i ia 1      = ∞→< > =      − ∑ − + ∞→ testno divergea converge P u u n i i n n n 1 &1 1 1lim 1 1
  • 75. 75 SOLO Convergence Tests Gauss’ s Test Carl Friedrich Gauss (1777 – 1855) If un > 0 for all finite n and in which B (n) is a bounded function of n for n → ∞, then converges for h > 1 and diverges for h ≤ 1. There is no indeterminate case here. ( ) 2 1 1 n nB n h u u n n ++= + ∑n nu Proof: For h > 1 and h < 1 the proof follows directly from Raabe’s Test: ( ) ( ) h n nB h n nB n h n u u n nn n n n =    +=    −++⋅=      −⋅ ∞→∞→ + ∞→ lim11lim1lim 2 1 If h = 1, Raabe’s Test fails. However if we return to Kummer’s Test and use an=n ln n: ( ) ( ) ( ) ( ) ( ) ( )             +−−+=    ++− + ⋅=       ++−    ++⋅ ∞→∞→ = ∞→ n nnnn n n nn nn n nB n h nn nn h n 1 1lnlnln1lim1ln1 1 lnlim 1ln11lnlim 1 2 Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series
  • 76. 76 SOLO Convergence Tests Gauss’ s Test Carl Friedrich Gauss (1777 – 1855) If un > 0 for all finite n and in which B (n) is a bounded function of n for n → ∞, then converges for h > 1 and diverges for h ≤ 1. There is no indeterminate case here. ( ) 2 1 1 n nB n h u u n n ++= + ∑n nu Proof (continue – 1): Kummer’s withan=n ln n: ( ) ( ) ( ) ( )             ++−=       ++−      ++⋅ ∞→ = ∞→ n nnn n nB n h nn n h n 1 1ln1lim1ln11lnlim 1 2 ( ) ( ) 01 3 1 2 11 1lim 1 1ln1lim 32 <−=      −+−⋅+−=      +⋅+− ∞→∞→  nnn n n n nn Hence we have a divergence for h = 1. This is an example of a successful application of Kremmer’s Test in which Raabe’s Test failed. Given a series: Theorems of Convergence of Sequences and Series ∑= = n i in uS 1 Infinite Series Return to the Table of Content
  • 77. 77 SOLO Complex Variables Infinite Series, Taylor’s and Laurent Series Let {un} :=u1 (z), u2 (z),…,un (z),…, be a sequence of single-valued functions of z in some region of z plane. We call U (z) the limit of {un} ,if given any positive number ε we can find a number N (ε,z) such that and we write this:( ) ( ) ( )zNnzUzun ,εε >∀<− ( ) ( ) ( ) ( )zUzuorzUzu n nn n ∞→ ∞→ →=lim x y C R If a sequence converges for all values z in a region R, we call R the region of convergence of the sequence. A sequence that is not convergent at some point z is called divergent at z. Infinite Series of Functions
  • 78. 78 SOLO Complex Variables Infinite Series, Taylor’s and Laurent Series Infinite Series of Functions From the sequence of functions {un} let form a new sequence {Sn} defined by: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑= =+++= += = n i inn zuzuzuzuzS zuzuzS zuzS 1 21 212 11   If , the series is called convergent and S (z) is its sum.( ) ( )zSzSnn =∞→ lim A necessary (but not sufficient) condition for convergence is that lim n→∞ un(z) = 0 Example: The Harmonic Series  ++++++=∑ ∞ = nnn 1 4 1 3 1 2 1 1 1 1 0 1 limlim == ∞→∞→ n u n n n By grouping the terms in the sum as ∞→+      + ++ + + + ++      ++++      +++ =>>>         2 1 22 1 2 1 1 2 1 1 1 8 1 7 1 6 1 5 1 4 1 3 1 2 1 1 p p pppp Return to the Table of Content
  • 79. 79 SOLO Complex Variables Absolute Convergence of Series of Functions Given a series of functions: ( ) ( )∑= = n i in zuzS 1 If is convergent the series is called absolutely convergent.( )∑= n i i zu 1 If is convergent but is not, the series is called conditionally convergent. ( )∑= n i i zu 1 ( )∑= n i i zu 1 Return to the Table of Content
  • 80. 80 SOLO Complex Variables Uniformly Convergence of Sequences and Series If for the sequence of functions {un(z)} we can find for each ε>0 a number N (ε) such that for all z∈R we say that {un} uniformly converges to U (z). ( N is a function only of ε and not of z) ( ) ( ) ( )εε NnzUzun >∀<− If the series of functions {Sn(z)} converges to S (z) for all z∈R we define the remainder ( ) ( ) ( ) ( )∑ ∞ += =−= 1 : nz inn zuzSzSzR The series of functions {Sn(z)} is uniformly convergent to S (z) if for all for all ε>0 and for all z∈R we can find a number N (ε) such that ( ) ( ) ( )εε NnzSzSn >∀<− x y C R Return to the Table of Content
  • 81. 81 SOLO Complex Variables Weierstrass M (Majorant) Test Karl Theodor Wilhelm Weierstrass (1815 – 11897) The most commonly encountered test for Uniform Convergence is the Weierstrass M Test. Proof: Since converges, some number N exists such that for n + 1 ≥ N, If we can construct a series of numbers , in which Mi ≥ |ui(x)| for all x in the interval [a,b] and is convergent, the series ui(x) will be uniformly convergent in [a,b]. ∑ ∞ 1 iM ∑ ∞ 1 iM ∑ ∞ 1 iM ε<∑ ∞ += 1ni iM This follows from our definition of convergence. Then, with |ui(x)| ≤ Mi for all x in the interval a ≤ x ≤ b, ( ) ε<∑ ∞ += 1ni i xu Hence ( ) ( ) ( ) ε<=− ∑ ∞ += 1ni in xuxsxS and by definition is uniformly convergent in [a,b]. Since we specified absolute values in the statement of the Weierstrass M Test, the series is also Absolutely Convergent. Return to the Table of Content ∑ ∞ =1i iu ∑ ∞ =1i iu
  • 82. SOLO Complex Variables Abel’s Test Niels Henrik Abel ( 1802 – 1829) If and the functions fn(x) are monotonic decreasing | fn+1(x) ≤ fn(x)| and bounded, 0 ≤ fn(x) ≤ M, for all x in [a,b], then Converges Uniformly in [a,b]. ( ) ( ) convergentAa xfaxu n nnn ,= = ∑ ( )∑n n xu ( ) ( ) [ ] ( ) [ ]bainconvergentuniformlyisxu xd d baincontinuousarexu xd d andxu n nnn ,&, 1∑ ∞ = Return to the Table of Content Uniformly convergent series have three particular useful properties: 1.If the individual terms un(x) are continuous, the series sum is also continuous. 2. If the individual terms un(x) are continuous, the series may be integrated term by term. The sum of the integrals is equal to to the integral of the sum, then 3.The derivative of the series sum f (x) equals the sum of the individual term derivatives providing the following conditions are satisfied ( ) ( )∑ ∞ = = 1n n xuxf ( ) ( )∑ ∫∫ ∞ = = 1n b a n b a xdxuxdxf ( ) ( )∑ ∞ = = 1n n xu xd d xf xd d
  • 83. SOLO Complex Variables Uniformly Convergent Series of Analytic Functions Suppose that (i)Each number of a sequence of functions u1(z), u2(z),…,un(z),… is Analytic inside a Region D, (ii)The Series is Uniformly Convergent through Every Region D’ interior to D. Then the function is Analytic inside D, and all its Derivatives can be calculated by term-by-term Differentiation. ( )∑ ∞ =1n n zu ( ) ( )∑ ∞ = = 1n n zuzf Proof: Let C be a simple closed contour entirely inside D, and let z a Point inside D. Since un(z) is Analytic inside D, we have: ( ) ( ) ∫ − = C n n wd zw wu i zu π2 1 for each function un(z). Hence ( ) ( ) ( ) ∑ ∫∑ ∞ = ∞ = − == 11 2 1 n C n n n wd zw wu i zuzf π
  • 84. SOLO Complex Variables Uniformly Convergent Series of Analytic Functions Proof (continue – 1): Since is Uniformly Convergent on C, we may multiply by 1/(w-z) and integrate term-by-term: and we obtain ( ) ( ) ( ) ∑ ∫∑ ∞ = ∞ = − == 11 2 1 n C n n n wd zw wu i zuzf π ( )∑ ∞ =1n n zu ( ) ( ) ∑ ∫∫∑ ∞ = ∞ = − = − 11 n C n C n n wd zw wu wd zw wu ( ) ( ) ( ) ∫∫∑ − = − = ∞ = CC n n wd zw wf i wd zw wu i zf ππ 2 1 2 1 1 The last integral proves that f(z) is Analytic inside C, and since C is an arbitrary closed contour inside D, f(z) is Analytic inside D.
  • 85. SOLO Complex Variables Uniformly Convergent Series of Analytic Functions Proof (continue – 2): Since f(z) is Analytic in D, the same is true for f’(z), therefore we can write ( ) ( ) ( )∫ − = C wd zw wf i zf 2 2 1 ' π Therefore q.e.d. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑∑ ∫ ∫∑∫ ∞ = ∞ = ∞ = = − = − = − = 11 2 1 22 ' 2 1 2 1 2 1 ' n nn C n eConvergenc Uniform C n n C zuwd zw wu i zw wd wu i wd zw wf i zf π ππ Hence the Series can be Differentiate term-by-term
  • 86. SOLO Complex Variables Uniformly Convergent Series of Analytic Functions Remarks on the above Theorem (i)The contrast between the conditions for term-by-term differentiation of Real Series, and of Series of Analytic Functions is that - In the case of Real Series we have to assume that the Differentiated Series is Uniformly Convergent. - In the case of Series of Analytic Series the Theorem proved that the Differentiated Series is Uniformly Convergent. (ii)If we merely assumed that the given Series is Uniformly Convergent on a certain Closed Curve C, we could prove as before that f(z) is Analytic at all points inside C. (iii) Even if we assume that each un(z) is Analytic on the Boundary of the Domain D, and the Series is Uniformly Convergent on the Boundary, we can not prove that f(z) is Analytic on the Boundary, or the Differentiated Series Converges on the Boundary. (iv) The Theorem may be stated as a Theorem on Sequences of Functions: If fn(z) is Analytic in D for each value of n, and tends to f(z) Uniformly in any Region interior to D, then f(z) is Analytic inside D, and fn’(z) tends to f’(z) Uniformly in any Region interior to D. Return to the Table of Content
  • 87. 87 SOLO Complex Variables Let f (z) be analytic at all points within a circle C0 with center at z0 and radius r0. Then at each point z inside C0: Taylor’s Series ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  +−++−+−+= n n zz n zf zz zf zzzfzfzf 0 02 0 0 000 !!2 '' ' Power Series Brook Taylor 1685 - 1731 a convergent power series for some |z-z0|<R (radius of convergence). C x y R z0 C0 C1 z z' r0 r1 r Proof: Start with the Cauchy’s Integral Formula: ( ) ( ) ∫ − = C zd zz zf i zf ' ' ' 2 1 π Use the identity: α α ααα α − +++++≡ − − 1 1 1 1 12 n n  for: ( ) ( )                     − − − − − +        − − +++ − − + − = − − − − = −+− = − − nn zz zz zz zzzz zz zz zz zz zz zzzzzzzzzz 0 0 0 0 1 0 0 0 0 0 0 0000 ' ' 1 1 '' 1 ' 1 ' 1 1 ' 1 ' 1 ' 1  Since z inside C0 |z-z0|=r < r0. For z’ is on C1 we have |z’-z0|=r1<r0
  • 88. 88 SOLO Complex Variables Taylor’s Series (continue - 1) Power Series C x y R z0 C0 C1 z z' r0 r1 r Proof (continue - 1): Using the Cauchy’s Integral Formula: ( ) ( ) ∫ − = C zd zz zf i zf ' ' ' 2 1 π ( ) ( )                     − − − − − +        − − +++ − − + − = − − nn zz zz zz zzzz zz zz zz zz zf zz zf 0 0 0 0 1 0 0 0 0 0 ' ' 1 1 '' 1 ' ' ' '  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n R C n n n nzf C n zf C zf C Rzz n zf zzzfzf zzzz zdzf i zz zz zz zdzf i zz zz zdzf izz zdzf i n n +−++−+= −− − + −         − ++−         − + − = ∫ ∫∫∫ − 0 0 000 0 0 1 0 !/ 0 0 !1/' 2 00 ! ' '' '' 2 ' '' 2 1 ' '' 2 1 ' '' 2 1 0 0 0 0 0 0 0              π πππ We have: ( ) ( ) n i n n C n n n r r rr Mr der rrr Mr zzzz zdzfzz R       − = − ≤ −− − ≤ ∫∫ 11 1 2 0 1 110 0 2'' '' 2 0 π θ θ ππ where |f (z)|<M in C0 and r/r1< 1, therefore: 0 ∞→ → n nR q.e.d. 0100 ' rrzzrzz <=−<=−
  • 89. 89 SOLO Complex Variables Let f (z) be analytic at all points within a circle C0 with center at z0 and radius r0. Then at each point z inside C0: Taylor’s Series (continue – 2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  +−++−+−+= n n zz n zf zz zf zzzfzfzf 0 02 0 0 000 !!2 '' ' Power Series Brook Taylor 1685 - 1731 a convergent power series for some |z-z0|<R (radius of convergence). C x y R z0 C0 C1 z z' r0 r1 r Proof (continue – 2): ( ) ( ) ( ) ( )∑ ∞ = −= 0 01 0 1 !k k k zz k zf zfSuppose the series converges for z=z1: ( ) ( ) ( ) ( ) ( )         − − − =≤         − − −=−≤ <∞ = ∞ = ∞ = ∑∑∑ 01 0 1 001 0 0 01 0 0 0 0 1 !! zz zz M aM zz zz zz k zf zz k zf zf a k k k k k k k k k Since the series converges all its terms are bounded ( ) ( ) ,2,1,0 ! 01 0 =∀<− nMzz k zf k k Define: 01 0 : zz zz a − − = Therefore the series f (z) converges for all 010 zzzz −<− The region of convergence of a Taylor series of f (z) around a point z0 is a circle centered at z and radius of convergence R that extends until f (z) stops to be analytic.
  • 90. 90 SOLO Complex Variables Taylor’s Series (continue – 3) ( ) ( ) ( ) ( ) ( ) ( )  +++++= n n z n f z f zffzf ! 0 !2 0'' 0'0 2 Power Series Brook Taylor 1685 - 1731 When z0 = 0 the series is called Maclaurin’s series after Colin Maclaurin a contemporary of Brook Taylor. Colin Maclaurin 1698 - 1746 Examples of Taylor’s Series ∞<= ∑ ∞ = z n z e n n z 0 ! ( ) ( ) ∞< − −= ∑ ∞ = − + z n z z n n n 0 12 1 !12 1sin ( ) ( ) ∞<−= ∑ ∞ = z n z z n n n 0 2 !2 1cos ( ) ∞< − = ∑ ∞ = − z n z z n n 0 12 !12 sinh ( ) ∞<= ∑ ∞ = z n z z n n 0 2 !2 cosh ( ) 11 1 1 0 <−= + ∑ ∞ = zz z n nn Return to the Table of Content
  • 91. 91 SOLO Complex Variables Laurent’s Series (1843) Power Series If f (z) is analytic inside and on the boundary of the ring shaped region R bounded by two concentric circles C1 and C2 with center at z0 and respective radii r1 and r2 (r1 > r2), then for all z in R: Pierre Alphonse Laurent 1813 - 1854 C1 x y R C2R2 R1 z0 z z' r P1 P0 z'( ) ( ) ( )∑∑ ∞ = − ∞ = − +−= 1 00 0 n n n n n n zz a zzazf ( ) ( ) ,2,1,0' ' ' 2 1 2 1 0 = − = ∫ +−− nzd zz zf i a C nn π ( ) ( ) ,2,1,0' ' ' 2 1 1 1 0 = − = ∫ + nzd zz zf i a C nn π Proof: Since z is inside R we have R1 <|z-z0|=r < R2 , and |z’-z0|= R1 on C1 and R2 on C2. Start with the Cauchy’s Integral Formula: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫∫∫∫∫∫ − − − =→ − + − + − + − = 212 0 1 1 01 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 0 CCC P P P PC dz zz zf dz zz zf zfdzdz zz zf dz zz zf dz zz zf dz zz zf zf   
  • 92. 92 SOLO Complex Variables Laurent’s Series (continue - 1) Power Series Pierre Alphonse Laurent 1813 - 1854 C1 x y R C2R2 R1 z0 z z' r Proof (continue – 1): Since z and z’ are inside R we have R1 >|z-z0|=r >R2, |z’-z0|=R1. From Cauchy’s Integral Formula: ( ) ( ) ( ) ∫∫ − − − = 21 ' ' ' ' ' ' CC dz zz zf dz zz zf zf Use the identity: α α ααα α − +++++≡ − − 1 1 1 1 12 n n  For I integral:                     − − − − − +        − − ++ − − + − = − − − − = − − nn zz zz zz zzzz zz zz zz zz zz zzzzzz 0 0 0 0 1 0 0 0 0 0 0 00 ' ' 1 1 '' 1 ' 1 ' 1 1 ' 1 ' 1  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n R C n n n zs C n za C za C Rzzzazzzaza zzzz zdzf i zz zz zz zdzf i zz zz zdzf izz zdzf i n n +−⋅++−⋅+= −− − + −         − ++−         − + − = ∫ ∫∫∫ − 0000100 0 0 1 0 0 02 00 2 0 2 01 2 00 2 '' '' 2 ' '' 2 1 ' '' 2 1 ' '' 2 1              π πππ ( ) ∫ −1 ' ' ' 2 1 C zd zz zf iπ We have: ( ) ( ) n n n C n n n R r rR MR dR rRR Mr zzzz zdzfzz R       − = − ≤ −− − ≤ ∫∫ 11 1 2 0 1 110 0 2'' '' 2 0 π θ ππ where |f (z)|<M in R and r/R1< 1, therefore: 0 ∞→ → n nR
  • 93. 93 SOLO Complex Variables Laurent’s Series (continue - 2) Power Series Pierre Alphonse Laurent 1813 - 1854 C1 x y R C2R2 R1 z0 z z' r Proof (continue – 1): Since z and z’ are inside R we have R1 >|z-z0|=r > R2, |z’-z0|=R2. From Cauchy’s Integral Formula: ( ) ( ) ( ) ∫∫ − − − = 21 ' ' ' ' ' ' CC dz zz zf dz zz zf zf Use the identity: α α ααα α − +++++≡ − − 1 1 1 1 12 n n  For II integral:                     − − − − − +        − − ++ − − + − = − − − − = − − − nn zz zz zz zzzz zz zz zz zz zz zzzzzz 0 0 0 0 1 0 0 0 0 0 0 00 ' ' 1 1'' 1 1 ' 1 11 ' 1  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n R C n n n za C n za CC Rzzzazzza zzzz zdzfzz i zzzz zdzf izzzz zdzf i zdzf i n n − +− +− − − +−+−− +−++−= −− − + −        − ++ −        − += − +−− ∫ ∫∫∫ 1 001 1 001 0 0 1 0 1 00 2 0 0 0 01 0 01 00 ' ''' 2 1 1 ' '' 2 11 ' '' 2 1 '' 2 1              π πππ ( ) ∫ −C zd zz zf i ' ' ' 2 1 π We have: ( ) ( ) n n n C n n n r R rR RM dR rRr MR zzzz zdzfzz R       − = − ≤ −− − ≤ ∫∫− 2 2 2 2 0 2 2 2 0 0 2' ''' 2 1 0 π θ ππ where |f (z)|<M in R and R2/r< 1, therefore: 0 ∞→ − → n nR Return to the Table of Content
  • 94. 94 SOLO Complex Variables Zeros of Holomorphic Functions ( ) ( ) ( ) ( ) ( ) ( ) ( ) 00 00 1 0 1 0 ≠==== − zfandzfzfzf kk  We say that Holomorphic Function f (z) has a Zero of Order k at z = z0 if If f (z) has a Zero of Order k at z = z0, by Taylor expansion, we can write with Holomorphic and nonzero. ( ) ( ) ( )zfzzzf k k 0−= ( ) ( ) ( )kk zz zf zf 0 : − = Note: (1) For g (z) = 1/ f (z) the Order k Zeros of f (z) are Order k Poles of g (z) ( ) ( ) ( ) ( )zfzzzf zg k k 111 0− == (2) For ( ) ( ) ( ) ( ) ( )zf zf zz k zf zf zf zd d k k '' ln 0 + − == z = z0, is a Simple Pole. Return to the Table of Content
  • 95. 95 SOLO Complex Variables Theorem: f(z) Analytic and Nonzero → ln|f(z)| Harmonic If f (z) is analytic for in an Open Set Ω, and has no zeros in Ω, then ln |f(z)| is Harmonic in Ω. Proof : Since f (z) is analytic and has no zeros the logarithm of f(z) is also Analytic g (z) := ln f (z) is Analytic Therefore q.e.d. ( ) ( ) ( ) ( ) ( ) ( )zgizgzgizgzg eeeezf ImReImRe === + and ( ) Harmoniczgizgzg )(Im)(Re += ( ) ( ) ( ) ( )zgzgizg eeezf Re 1 ImRe =⋅=  ( ) ( ) ( ) Harmoniczgezf zg Relnln Re == meaning Harmonicyixgyixg yixg y yixg x yixg y yixg x )(Re),(Re 0)(Im)(Im&0)(Re)(Re 2 2 2 2 2 2 2 2 ++⇔ =+ ∂ ∂ ++ ∂ ∂ =+ ∂ ∂ ++ ∂ ∂ Return to the Table of Content
  • 96. 96 SOLO Complex Variables Polynomial Theorem If f (z) is analytic for all finite values of z, and as |z| → ∞, and then f (z) is a polynomial of degree ≤ k. Proof : Integrating this result we obtain q.e.d. ( ) kgivenAsomeforzforzAzf k &0, >∞→≤ Using Taylor Series around any analytic point z = a ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  +⋅ − ++⋅−+= af n az afazafzf n n ! 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Aafaz z az n af z az k af z az z af z zf nkn k k k k k kkk ≤+⋅−⋅ − ++⋅ − ++⋅ − +≤ −  ! 1 ! 11 If |z| → ∞ , the previous equation is possible only if f(n) (a)=0 for all n > k and all a. Therefore f (k) (a) = constant for all a, i.e. f (k) (z)=ak=constant. ( ) 0 1 1 azazazf k k k k +++= − −  Continuing to Integrate we obtain ( ) ( ) 1 1 − − += kk k azazf Return to the Table of Content
  • 97. 97 SOLO Complex Variables The Argument Theorem If f (z) is analytic inside and on a simple closed curve C except for a finite number of poles inside C (this is called a Meromorphic Function), then ( ) ( ) PNdz zf zf i C −=∫ ' 2 1 π where N and P are respectively the number of zeros and poles inside C. Proof: Let write f (z) as: ( ) ( ) ( ) ( )zG z z zf j p j k n k j k ∏ ∏ − − = α β where: ,∑∑ == j j k k pPnN & ( ) ( ) ( ) ( )zGzpznzf jjk k k lnlnlnln +−−−= ∑∑ αβ Differentiate this equation: ( ) ( ) ( ) ( ) ( ) ( ) Cinanalytic j j k k k zG zG z p z n zf zf ' ' + − − − = ∑∑ αβ ( ) ( ) ( ) ( ) ( ) ( ) PNpn zG zG iz p iz n i dz zf zf i j k k C p C j j k n C k k C jk −=−= + − − − = ∑∑ ∫∑∫∑ ∫∫        0 ' 2 1 2 1 2 1' 2 1 παπβππ and G (z) ≠ 0 and analytic in C (G’ (z) exists). ( ) ( ) k C k k C k k ndz z n i dz z n i k = − = − ∫∫ β βπβπ 2 1 2 1 ( ) ( ) j C j j C j j pdz z p i dz z p i j = − = − ∫∫ α απαπ 2 1 2 1 x y C R 1α 3 α 2α kα 1β 2β j β k Cα 3αC 2αC 2βC 1βC j Cβ q.e.d. Return to the Table of Content
  • 98. 98 SOLO Complex Variables Rouché’s Theorem Eugène Rouché 1832 - 1910 If f (z) and g (z) are analytic inside and on a simple closed curve C and if |g (z)| < |f (z)| on C, then f (z) + g (z) and f (z) have the same number of zeros inside C. Proof: Let F (z):= g (z)/f (z) If N1 and N2 are the number of zeros inside C of f (z) + g (z) and f (z) respectively, and using the fact that those functions are analytic and C, therefore they have no poles inside C, using the Argument Theorem we have ( ) ( )∫= C dz zf zf i N ' 2 1 2 π ( ) ( ) ( ) ( )∫ + + = C dz zgzf zgzf i N '' 2 1 1 π ( ) ( ) ( ) 0 1' 2 1 1 ' 2 1' 2 1 1 '' 2 1 ' 2 1 1 '1' 2 1' 2 1''' 2 1 32 21 = +−+−= + =−      + += − + ++ =− + ++ =− ∫∫∫∫ ∫∫∫∫ CCCC CCCC dzFFFF i dz F F i dz f f i dz F F f f i dz f f i dz Ff FfFf i dz f f i dz Fff FfFff i NN  ππππ ππππ We used the fact that |F|=|g/f|<1 on C, so the series 1-F+F2 +… is uniformly convergent on C and integration term by term yields the value zero. Thus N1=N2 q.e.d. Return to the Table of Content
  • 99. 99 SOLO Complex Variables Foundamental Theorem of Algebra (using Rouché’s Theorem) Every polynomial equation P (z) = a0 + a1z+a2z2 +…+anzn =0 with degree n ≥ 1 and an ≠ 0 (ai are complex constants) has at exactely n zeros. Proof: Define: Take C as the circle with the center at the origin and radius r > 1. q.e.d. ( ) n n zazf =: ( ) 1 1 2 210 : − − ++++= n n zazazaazg  ( ) ( ) ra aaaa ra rararara ra rararaa za zazazaa zf zg n n n n n n nnn n n n n n n n n 1210 1 1 1 2 1 1 1 0 1 1 2 210 1 1 2 210 − − − −−− − − − − ++++ = ++++ ≤ ++++ ≤ ++++ =   By choosing r large enough we can make |g (z)|/|f (z)|<1, and using Rouché’s Theorem ( ) ( ) ( ) n n n n zazazazaazgzfzP +++++=+= − − 1 1 2 210  and (n zeros at the origin z = 0) have the same number of zeros, i.e. P (z) has exactly n zeros. ( ) n n zazf =: Return to the Table of Content
  • 100. 100 SOLO Complex Variables Jensen’s Formula Johan Ludwig William Valdemar Jensen (1859 – 5 1925) which is the Mean-Value Property of the Harmonic Function ln |f(z)|. Suppose that ƒ is an Analytic Function in a region in the complex plane which contains the closed disk D of radius r about the origin, a1, a2, ..., an are the zeros of ƒ in the interior of D repeated according to multiplicity, and ƒ(0) ≠ 0. Jensen's formula states that This formula establishes a connection between the Moduli of the zeros of the function ƒ(z) inside the disk D and the average of log |f(z)| on the boundary circle |z| = r, and can be seen as a generalization of the Mean Value Property of Harmonic Functions. Namely, if f(z) has no zeros in D, then Jensen's formula reduces to ( ) ( )∫∑ +        = = π θ θ π 2 01 ln 2 1 ln0ln derf r a f j n k k ( ) ( )∫= π θ θ π 2 0 ln 2 1 0ln derff j
  • 101. 101 SOLO Complex Variables Jensen’s Formula (continue – 1) Proof If f has no Zeros in D, then we can use Gauss’ Main Value Theorem to ln f(z) that is Harmonic in D ( ) ( )∫= π θ θ π 2 0 ln 2 1 0ln derff j Since f has Zeros a1, a2,…, an inside D, ( |z| < r) let define the Holomorphic Function F (z) without Zeros in D : ( ) ( ) ( )∏= − − ⋅= n k k k raz rza zfzF 1 2 / /1 : Apply the Gauss’ Main Value theorem for |z|=r ( ) ( ) ( ) ( )∫∫∑ ==        −= = π θ π θ θ π θ π 2 0 2 01 ln 2 1 ln 2 1 ln0ln0ln derfderF r a fF ii n k k and:   1 1 1 1 /1 1 /1 / /1 / /1 2 2 1 2 22 1 22 22 = − − = − − = − − = − − = − − →=⋅= r z a r z a r zz z a rza z a rza z r raz rza raz rza rzzz k k k k k k k k k k The Zeros of f(z) are cancelled. q.e.d. We have: i.e. zk1 is outside the Disk D. rzarzforrza k ra kkkk k >→==− < 1 2 1 2 1 /0/1
  • 102. 102 SOLO Complex Variables Jensen’s Formula (continue – 2) Johan Ludwig William Valdemar Jensen (1859 – 5 1925) ( ) ( ) ( )zh zg zzf l = Jensen's formula may be generalized for functions which are merely meromorphic on D. Namely, assume that where g and h are analytic functions in D having zeros at respectively, then Jensen's formula for meromorphic functions states that Jensen's formula can be used to estimate the number of zeros of analytic function in a circle. Namely, if f is a function analytic in a disk of radius R centered at z0 and if |f(z)| is bounded by M on the boundary of that disk, then the number of zeros of f(z) in a circle of radius r < R centered at the same point z0 does not exceed { } { }0,,0,, 11 DbbandDaa mn ∈∈  ( ) ( ) ( )∫+= − π θ π 2 01 1 ln 2 1 ln 0 0 ln i m nnm erf bb aa r h g   ( ) ( )0 ln /ln 1 zf M rR
  • 103. 103 SOLO Complex Variables Jensen’s Formula (continue – 3) Johan Ludwig William Valdemar Jensen (1859 – 5 1925) Jensen's formula may be put in an other way. If n (r) denotes the number of Zeros, including multiplicity, and p (r) denotes the number of Poles, including multiplicity, for |z| < r, then the Jensen’s Formula can be written as ( ) ( ) ( ) ( )0lnln 2 1 lnln 2 0110 fderf b r a r dx x xpxn j n j j m k k r −=         −        = − ∫∑∑∫ == π θ θ π Proof         −−−=         −        ∑∑∑∑ ==== n j j m k k n j j m k k brnarm b r a r 1111 lnlnlnlnlnln ( ) ( ) ( ) ( )        −+−−−+−= ∑∑ − = + − = + n n j jjm m k kk brnbbjarnaak lnlnlnlnlnlnlnln 1 1 1 1 1 1         +−         += ∫∑ ∫∫∑ ∫ − = − = ++ r r n j r r r r m k r r n j jn k k x xd n x xd j x xd m x xd k 1 1 1 1 11
  • 104. 104 SOLO Complex Variables Jensen’s Formula (continue – 4) Johan Ludwig William Valdemar Jensen (1859 – 5 1925) Jensen's formula may be put in an other way. If n (r) denotes the number of Zeros, including multiplicity, and p (r) denotes the number of Poles, including multiplicity, for |z| < r, then the Jensen’s Formula can be written as ( ) ( ) ( ) ( )0lnln 2 1 lnln 2 0110 fderf b r a r dx x xpxn j n j j m k k r −=         −        = − ∫∑∑∫ == π θ θ π Proof (continue – 1)         +−         +=         −        ∫∑ ∫∫∑ ∫∑∑ − = − === ++ r r n j r r r r m k r r n j j m k k n j jn k k x xd n x xd j x xd m x xd k b r a r 1 1 1 111 11 lnln But k = n (x) for rm ≤ x ≤ rm+1, m = n (x) for rn ≤ x ≤ r, and j = p (x) for rj ≤ x ≤ rj+1, n = p (x) for rn ≤ x ≤ r Hence ( ) ( ) xd x xp xd x xn b r a r rrn j j m k k ∫∫∑∑ −=         −        == 0011 lnln q.e.d. Return to the Table of Content
  • 105. 105 SOLO Complex Variables Poisson-Jensen’s Formula for a Disk Johan Ludwig William Valdemar Jensen (1859 – 5 1925) Poisson Formula states: Siméon Denis Poisson 1781-1840 Let g (z) be analytic inside and on the circle C defined by |z| = R, and let z = r e iθ be any point inside C, then: ( ) ( )∫ − − = π φ θφ θ φ π 2 0 2 22 2 1 deRg ereR rR erg i ii i In our case ƒ is an analytic function in a region in the complex plane which contains the closed disk D of radius R about the origin, a1, a2, ..., am are the Zeros, and b1, b2, ..., bn are the Poles of ƒ in the interior of D repeated according to multiplicity. Since f has Zeros a1, a2,…, an inside D, ( |z| < r) let define the Holomorphic Function F (z) without Zeros in D : ( ) ( ) ( ) ( ) Rz Rzb Rbz Raz Rza zfzF m k n j j j k k = − − ⋅ − − ⋅= ∏ ∏= =1 1 2 2 /1 / / /1 : Zeros and Poles of f(z) are cancelled, and new ones are outside D.
  • 106. 106 SOLO Complex Variables Poisson-Jensen’s Formula for a Disk Johan Ludwig William Valdemar Jensen (1859 – 5 1925) Let apply the Poisson Formula to g (z) = ln |F (z)|: ( ) ( ) ( ) ( ) ( ) ∏∏ == − − + − − +== n j j j m k k k Rzb Rbz Raz Rza zfzFzg 1 2 1 2 /1 / ln / /1 lnlnln: ( ) ( ) 1 /1 / / /1 2 2 Rz j j Rz k k Rzb Rbz Raz Rza == = − − = − − Siméon Denis Poisson 1781-1840 We proved that: ( ) ( )φφ ii eRfeRg ln= The Poisson Formula to g (z) = ln F (z) is: ( ) ( ) ( ) ( ) RzdeRf ereR rR Rzb Rbz Raz Rza zf i ii n j j j m k k k < − − = − − + − − + ∫ ∏∏ == π φ θφ φ π 2 0 2 22 1 2 1 2 ln 2 1 /1 / ln / /1 lnln RzRzbandRzRza jjjkkk >→=−>→=− 1 2 11 2 1 0/1,0/1
  • 107. 107 SOLO Complex Variables Poisson-Jensen’s Formula for a Disk Johan Ludwig William Valdemar Jensen (1859 – 5 1925) Siméon Denis Poisson 1781-1840 The Poisson-Jensen’s Formula for a Disk is: ( ) ( ) ( ) ( ) RzdeRf ereR rR Rzb Rbz Raz Rza zf i ii n j j j m k k k < − − = − − + − − + ∫ ∑∑ == π φ θφ φ π 2 0 2 22 1 2 1 2 ln 2 1 /1 / ln / /1 lnln For z = r = 0 we obtain the Jensen’s Formula: ( ) ( )∫=       + − π φ φ π 2 021 21 ln 2 1 ln0ln deRfR aaa bbb f inm m n   If there are no Zeros or Poles in D, it reduces to Poisson’s Formula: ( ) ( ) RzdeRf ereR rR zf i ii < − − = ∫ π φ θφ φ π 2 0 2 22 ln 2 1 ln Return to the Table of Content
  • 108. 108 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series If f (z) is analytic inside and on the boundary of a circle C, except it’s center z0, then according to Laurent’s Series: C x y R R z0 z z' r P0 ( ) ( ) ( )∑∑ ∞ = − ∞ = − +−= 1 00 0 n n n n n n zz a zzazf ( ) ( ) ,2,1,0' ' ' 2 1 1 0 = − = ∫ +−− nzd zz zf i a C nn π ( ) ( ) ,2,1,0' ' ' 2 1 1 0 = − = ∫ + nzd zz zf i a C nn π Let compute ( ) ( ) ( ) ( ) ,2,1,0' ' ' '''' 1 00 0 = − +−= ∑ ∫∫ ∑ ∫ ∞ = − ∞ = nzd zz zf azdzzazdzf n C nn C n C n n ( ) ( ) ( )    = ≠ = − ==− ∫∫ 12 10 ' ' ' &,2,1,00'' 0 0 ni n zd zz zf nzdzz C n C n π  Therefore: ( ) 12'' −∫ = aizdzf C π Because only a-1 is involved in the integral above, it is called the residue of f (z) at z = z0. Return to the Table of Content
  • 109. 109 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series According to residue definition the residue of f (z) at z = z0 can be computed as follows: C x y R R z0 z z' r P0 If z = z0 is a pole of order k, i.e. the Laurent series at z0 is Then: ( )∫=− C zdzf i a '' 2 1 1 π Calculation of the Residues ( ) ( ) ( ) ( ) ( )  +−+−++ − ++ − + − = −−− 2 02010 0 2 0 2 0 1 zzazzaa zz a zz a zz a k k ( ) ( ) ( )[ ]zfzz zd d k a k k k zz 01 1 1 !1 1 lim 0 − − = − − →− ( ) ( ) ( ) ( ) ( ) ( ) ( )  +−+−+−+++−+−=− ++ − − − − − 2 02 1 0100 2 02 1 010 kkk k kkk zzazzazzaazzazzazfzz and: If z = z0 is a pole of order k=1, then: ( ) ( )[ ]zfzza zz 01 0 lim −= →− ( ) ( ) ( )∑∑ = − ∞ = − +−= k n n n n n n zz a zzazf 1 00 0 Return to the Table of Content
  • 110. 110 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series The Residue Theorem If f (z) is analytic inside and on the boundary of a closed curve C, except at the singularities z01, z02,…,z0n, which have residues Re1, Re2,…,Ren, then: Proof: ( ) ( )n C izdzf ReReRe2'' 21 +++=∫ π x y C R 01z nzC 2zC 02z 1zC nz0 Surround every singularity z0i by a small closed curve Czi, that enclosed only this singularity. Connect those Curves to C by a small corridor (the width of which shrinks to zero, so that the integration along the opposite directions will cancel out) ( ) ( ) ( ) ( ) 0'''''''' 21 =−−−− ∫∫∫∫ znzz CCCC zdzfzdzfzdzfzdzf  We have , therefore:( ) i C izdzf zi Re2'' π∫ = ( ) ( ) ( ) ( ) ( )n CCCC izdzfzdzfzdzfzdzf znzz ReReRe2'''''''' 21 21 +++=+++= ∫∫∫∫  π q.e.d. Return to the Table of Content
  • 111. 111 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Theorem 1 If |F (z)| ≤ M/Rk for z = R e iθ where k > 1 and M are constants, then where Γ is the semicircle arc of radius R, center at origin, in the upper part of z plane. ( ) 0lim =∫Γ →∞ zdzFR x y Γ R Proof: ( ) 1 0 1 0 == = ΓΓ =≤=≤ ∫∫∫∫ kk i k eRz k R M d R M deRi R M zd R M zdzF i π θθ ππ θ θ Therefore: ( ) 0limlim 1 1 0 1 > −→∞− Γ →∞ ==≤ ∫∫ k kRkR R M d R M zdzF π θ π ( ) 0lim =∫Γ →∞ zdzFR and: ( ) ( ) ( ) 0limlimlim0 =≤≤−= ∫∫∫ Γ →∞ Γ →∞ Γ →∞ zdzFzdzFzdzF RRR q.e.d.Return to the Table of Content
  • 112. 112 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Jordan’s Lemma If |F (z)| ≤ M/Rk for z = R e iθ where k > 0 and M are constants, then where Γ is the semicircle arc of radius R, center at origin, in the upper part of z plane, and m is a positive constant. ( ) 0lim =∫Γ →∞ zdzFe zmi R x y Γ R Proof: ( ) 0lim =∫Γ →∞ zdzFe zmi R using: q.e.d. ( ) ( )∫∫ = Γ = π θθ θ θ θ 0 deRieRFezdzFe iieRmi eRz zmi i i ( ) ( ) ( ) ( ) ∫∫∫ ∫∫∫ − − − − − − =≤= =≤ 2/ 0 sin 1 0 sin 1 0 sin 0 sincos 00 2 π θ π θ π θθ π θθθθ π θθ π θθ θθθ θθθ θθ dRe R M dRe R M dReRFe deRieRFedeRieRFedeRieRFe Rm k Rm k iRm iiRmRmiiieRmiiieRmi ii 2/0/2sin πθπθθ ≤≤≥ for π2/π 1 θsin πθ /2 θ ( ) ( )Rm k Rm k Rm k iieRmi e R M de R M de R M deRieRFe i −− − − − −=≤≤ ∫∫∫ 1 222 2/ 0 /2 1 2/ 0 sin 1 0 π π π θ π θθ θθθ θ ( ) ( ) 01 2 limlim 0 =−≤ − →∞→∞ ∫ Rm kR iieRmi R e R M deRieRFe i π θθ θ θ Marie Ennemond Camille Jordan 1838 - 1922
  • 113. 113 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Jordan’s Lemma Generalization If |F (z)| ≤ M/Rk for z = R e iθ where k > 0 and M are constants, then for Γ a semicircle arc of radius R, and center at origin: ( ) 00lim >=∫Γ →∞ mzdzFe zmi R x y Γ R where Γ is the semicircle, in the upper part of z plane. 1 ( ) 00lim <=∫Γ →∞ mzdzFe zmi R x y Γ R where Γ is the semicircle, in the down part of z plane. 2 ( ) 00lim >=∫Γ →∞ mzdzFe zm R x y Γ R where Γ is the semicircle, in the right part of z plane. 3 ( ) 00lim <=∫Γ →∞ mzdzFe zm R where Γ is the semicircle, in the left part of z plane. 4 x yΓ R Return to the Table of Content
  • 114. 114 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Integral of the Type Bromwwich-Wagner ( )∫ ∞+ ∞− jc jc ts sdsFe iπ2 1 The contour from c - i ∞ to c + i ∞ is called Bromwich Contour Thomas Bromwich 1875 - 1929 x y 0< Γt R c x y 0>Γt R c ( ) ( ) ( ) ( ) ( ) ( ) ( )    < > ==         +== ∫ ∫∫∫ Γ ∞+ ∞− →∞ ∞+ ∞− 0 0 2 1 lim 2 1 2 1 tzFeRes tzFeRes zdzF i sdsFesdsFe i sdsFe i tf tz planezRight tz planezLeft ts ic ic ts R ic ic ts π ππ where Γ is the semicircle, in the right part of z plane, for t < 0. where Γ is the semicircle, in the left part of z plane, for t > 0. This integral is also the Inverse Laplace Transform. Return to the Table of Content
  • 115. 115 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Integral of the Type ,F (sin θ, cos θ) is a rational function of sin θ and cos θ ( )∫ π θθθ 2 0 cos,sin dF Let z = e iθ 22 cos, 22 sin 11 −−−− + = + = − = − = zzee i zz i ee iiii θθθθ θθ zizdddzideizd i /=→== θθθθ ( ) ∫∫       +− = −− C zi zdzz i zz FdF 2 , 2 cos,sin 112 0 π θθθ where C is the unit circle with center at the origin. C x y R=1 Return to the Table of Content
  • 116. 116 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Definite Integrals of the Type .( )∫ +∞ ∞− xdxF If the conditions of Theorem 1, i.e.: if |F (z)| ≤ M/Rk for z = R e iθ where k > 1 and M are constants, then and we can write ( ) 0lim =∫Γ →∞ zdzFR x y Γ R ( ) ( ) ( ) ( ) ( )zFResizdzFzdzFxdxFxdxF planezUpper R R R π2lim ==         += ∫∫∫∫ Γ + − →∞ +∞ ∞− Example: Heaviside Step Function ( ) x e i xF txi π2 1 := x y Γ R 0>t x y Γ R 0<t This function has a single pole at z = 0. For t > 0 Γ is the semicircle, in the upper part of z plane. We also include on the path x = - ∞ to x = + ∞ a small semicircle such that the pole z = 0 is included. For t < 0 Γ is the semicircle, in the lower part of z plane. We also include on the path x = - ∞ to x = + ∞ a small semicircle such that the pole z = 0 is excluded.     <= >= =∫ +∞ ∞− 00/ 01/ 2 1 tzeRes tzeRes xd x e i tzi planezLower tzi planezUpper txi π Return to the Table of Content
  • 117. 117 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Cauchy’s Principal Value Cauchy’s Principal value deals with integrals that have singularities along the integration paths. Start with the following: Theorem 1: If f (z) is analytic on and inside a positive-sensed circle C of radius ε, centered at z = z0, then ( ) ( )0 0 0 lim zfi zz zdzf C ψ ψ ε = −∫→ where Cψ is every arc on C of angle ψ. Proof: Since f (z) is analytic inside and on C we can use the Taylor series expansion to write ( ) ( ) ( ) ( ) ( )∑ ∞ = −+= 1 0 0 0 !n n n zz n zf zfzf ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )    zg n n n zz n zf zz zf zz zf ∑ ∞ = − −+ − = − 1 1 0 0 0 0 0 ! Consider the integral on Cψ defined by z=z0 + ε e iθ θ0 ≤ θ ≤ θ0 + ψ C x y ψC 0 z 0 θ 0 θψ + ψ ε O
  • 118. 118 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Cauchy’s Principal Value (continue – 1) Proof (continue – 1): Since g (z) is bounded inside and on C , there is a positive number M such that |g (z)| < M for all z such that |z – z0| < ε ( ) ( ) ( )∫∫∫ + − = − ψψψ CCC zdzg zz zdzf zz zdzf 0 0 0 ( ) ( ) ( ) ( )00 0 0 0 0 0 0 0 zfidizf zz zd zf zz zdzf i ezz CC ψθ ψθ θ ε θ ψψ == − = − ∫∫∫ + += ( ) ( ) ( )εψ ψψ MLMzdzgzdzg CC =<≤ ∫∫ Where L = ψ ε is the length of Cψ. ( ) ( ) ( ) 0lim0limlim 000 =⇒=≤ ∫∫ →→→ ψψ εεε εψ CC zdzgMzdzg ( ) ( ) ( ) ( ) 0lim0limlimlim0 0000 =⇒=≤≤−= ∫∫∫∫ →→→→ ψψψψ εεεε CCCC zdzgzdzgzdzgzdzg Therefore ( ) ( )0 0 0 lim zfi zz zdzf C ψ ψ ε = −∫→ q.e.d. Note: For ψ = 2 π we recover the Cauchy’s Integral result. C x y ψC 0 z 0 θ 0θψ + ψ ε O
  • 119. 119 SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Cauchy’s Principal Value (continue – 2) Theorem 2: If F (z) is analytic on and inside a positive-sensed circle C of radius ε, except at the center of C, z = z0, that is a simple pole of F (z), then C x y ψC 0 z 0 θ 0 θψ + ψ ε O ( ) ( )[ ]0 0 lim zFResizdzF C ψ ψ ε =∫→ where Cψ is every arc on C of angle ψ. Proof: Since f (z) is analytic inside and on C by using Theorem 1 we obtain the desired result. The function: ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]    =−= ≠− = → 000 00 0 lim zzzFzzzFRes zzzFzz zf zz

Editor's Notes

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  5. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Tartaglia_v_Cardan.html http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Quadratic_etc_equations.html http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cardan.html http://en.wikipedia.org/wiki/Cubic_equation
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