JITENDRA SUWASIYA, profile picture
Uploaded byJITENDRA SUWASIYA
1,769 views

Complex analysis book by iit

This document is the preface and contents for a set of notes on complex analysis. It was prepared by Charudatt Kadolkar for MSc students at IIT Guwahati in 2000 and 2001. The notes cover topics such as complex numbers, functions of complex variables, analytic functions, integrals, series, and the theory of residues and its applications. The contents section provides an outline of the chapters and topics to be covered.

Engineering◦
In this document
Powered by AI

Introduction to Complex Analysis course, acknowledgments, and course contents covering basic definitions, properties, and function types.

Detailed definitions and properties of complex numbers including algebraic operations, polar coordinates, Euler's formula, and roots in the complex plane.

Introduction to complex variable functions, elementary functions, mappings, and logarithmic functions with notable mappings by elementary functions.

Concepts of limits, continuity, derivatives, Cauchy-Riemann equations, and analytic functions with examples and characteristics.

Integrals in complex analysis, contour definitions, Cauchy-Goursat theorem, and methods for computing contour integrals.

Convergence of sequences and series, Taylor series, and Laurent series with definitions and examples.

Introduction to residues, singularities, types of singularities, and their applications in complex analysis with examples.

Appendices including references and the index for further reading and navigation in the course material.

Embed presentation

Downloaded 34 times
Complex Analysis PH 503 CourseTM Charudatt Kadolkar Indian Institute of Technology, Guwahati
ii Copyright   ¡ 2000 by Charudatt Kadolkar
Preface Preface Head These notes were prepared during the lectures given to MSc students at IIT Guwahati, July 2000 and 2001.. Acknowledgments As of now none but myself IIT Guwahati Charudatt Kadolkar.
Contents Preface iii Preface Head iii Acknowledgments iii 1 Complex Numbers 1 De• nitions 1 Algebraic Properties 1 Polar Coordinates and Euler Formula 2 Roots of Complex Numbers 3 Regions in Complex Plane 3 2 Functions of Complex Variables 5 Functions of a Complex Variable 5 Elementary Functions 5 Mappings 7 Mappings by Elementary Functions. 8 3 Analytic Functions 11 Limits 11 Continuity 12 Derivative 12 Cauchy-Riemann Equations 13
vi Contents Analytic Functions 14 Harmonic Functions 14 4 Integrals 15 Contours 15 Contour Integral 16 Cauchy-Goursat Theorem 17 Antiderivative 17 Cauchy Integral Formula 18 5 Series 19 Convergence of Sequences and Series 19 Taylor Series 20 Laurent Series 20 6 Theory of Residues And Its Applications 23 Singularities 23 Types of singularities 23 Residues 24 Residues of Poles 24 Quotients of Analytic Functions 25 A References 27 B Index 29
1 Complex Numbers De• nitions De• nition 1.1 Complex numbers are de•ned as ordered pairs ¢ £ ¤ ¥ ¦ Points on a complex plane. Real axis, imaginary axis, purely imaginary numbers. Real and imaginary parts of complex number. Equality of two complex numbers. De• nition 1.2 The sum and product of two complex numbers are de•ned as follows: § ¨ © © ! # $ % ' # $ ' % ( ! # ' # ( ) 0 1 2 3 4 2 5 6 0 1 7 1 2 8 9 @ 9 A B C @ 9 A D C A 9 @ E In the rest of the chapter use F G F H G F I G P P P for complex numbers and Q R S for real numbers. introduce T and U V W X T Y notation. Algebraic Properties 1. Commutativity U ` X U a V U a X U ` b U ` U a V U a U ` 2. Associativity c U ` X U a d X U e f g h i p g q i g e r s p g h g q r g e f g h p g q g e r 3. Distributive Law g p g h i g q r f g g h i g g q 4. Additive and Multiplicative Indentity g i t u v w v x y €  5. Additive and Multiplicative Inverse ‚  € ƒ ‚ „ … ‚ † ‡  ˆ ‰  ‘ ’ “ ” • – ” — ˜ – “ ” • – ” ™ d e f g h
2 Chapter 1 Complex Numbers 6. Subtraction and Division i j k i l m i j n o k i l p q i j i l m i j i r j l 7. Modulus or Absolute Value s t s u v w x y z x 8. Conjugates and properties { | w } ~ z |  w € } z  { ‚ ƒ „ … † „ ‡ ƒ „ … „ ‡ „ … † „ ‡ „ … ˆ „ ‡ „ … ‰ † „ ‡ „ … 9. Š „ Š … † „ „ ‹ Œ  Ž     ‘ ’ “ ” • ” – ” — ˜ 10. Triangle Inequality ™ ” š › ” œ ™  ž Ÿ   ž ¡ ž Ÿ ¢ ž Polar Coordinates and Euler Formula 1. Polar Form: for Ÿ £ ¤ ¥ , Ÿ ¤ ¦ § ¨ © ª « ¬ ­ ® ¯ ° ± ² where ³ ´ µ ¶ µ and · ¸ ¹ º » ¼ ½ ¾ . ¿ is called the argument of À . Since ¿ Á Â Ã Ä is also an argument of Å Æ the principle value of argument of Å is take such that Ç Ä È É Ê Ë Ì For Í Î Ï the Ð Ñ Ò Ó is unde• ned. 2. Euler formula: Symbollically, Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ Û ß à Ü á 3. â ã â ä × å ã å ä æ ç è é ê ë ì í î ï ð ï ñ ò ó ð ó ñ ô õ ö ÷ ø ù ÷ ú î ï û ü ý û þ û ÿ 4. de Moivre’s Formula   ¡ ¢ £ ¤ ¥ ¦ £ § ¨ ¤ © û ü ¡ ¢ £ ¤ ¥ ¦ £ § ¨ ¤
Roots of Complex Numbers 3 Roots of Complex Numbers Let then ! # $ % ' ( ) 0 1 2 34 5 5 There are only 4 distinct roots which can be given by 2 6 7 8 9 8 @ @ @ 8 4 A 9 @ If B is a principle value of C D E F then B G 4 is called the principle root. Example 1.1 The three possible roots of H I P Q R S T U V W X Y ` a b c d e f g h are i p q g f r s i p q g f r t p r q g h s i p q g f r t p u q g h . Regions in Complex Plane 1. v w x y € of  ‚ is de• ned as a set of all points ƒ which satisfy „ ƒ … ƒ ‚ „ † ‡ 2. ˆ ‰  ‘ ’ “ ” • – — of ˜ ™ is a nbd of ˜ ™ excluding point ˜ ™ . 3. Interior Point, Exterior Point, Boundary Point, Open set and closed set. 4. Domain, Region, Bounded sets, Limit Points.
2 Functions of Complex Vari- ables Functions of a Complex Variable A d e f g h i j k l de• ned on a set m is a rule that uniquely associates to each point n of m a a complex number o p Set q is called the r s t u v w of x and y is called the value of x at z and is denoted by x { z | } y ~ x { z | } x {  €  ‚ | } ƒ „ … † ‡ ˆ ‰ Š ‹ Œ  Ž   ‘ Œ ’  “ ‘ Œ ” • – —  “ ˜ Œ ” Ž ™  š › ‹ Œ ” Ž ™  “ œ  ž Ÿ   ¡ ¢ £ ¤ ¥ ¦ § ¨ © Example 2.1 Write ª « ¬ ­ ® ¯ ° ¬ ± in ² ³ ´ µ form. ² « ¶ · ¸ ­ ® ¹ º » ¼ ½ ¾ ¿ ½ À ¼ ½ Á ½ and Â Ã Ä Å Æ Ç È É ¿ ¼ ¾ ¿ ½ À ¼ ½ Á ½ Ê Ã Ë Å Ì Ç È Ë Í É Î Ï Ð Ñ Ì and Â Ã Ë Å Ì Ç È Ò Ë Í É Ð Ó Ô Ñ Ì Domain of Õ is Ö × Ø Ù Ú . A Û Ü Ý Þ ß à á â ã ä å æ ç è é ê ç ë ì í î ï ë is a rule that assigns more than one value to each point of domain. Example 2.2 ð ñ ò ó ô õ ö ÷ This function assigns two distinct values to each ö ø ù ú û ü . One can choose the function to be single-valued by specifying õ ö ú ý õ þ ÿ   ¡ ¢ £ where ¤ is the principal value. Elementary Functions
6 Chapter 2 Functions of Complex Variables 1. Polynomials ¥ ¦ § ¨ © whrere the coef• cients are real. Rational Functions. 2. Exponential Function ! # # $ % ' ( ) 0 1 2 3 3 3 Converges for all 4 5 For real 4 the de• nition coincides with usual exponential func- tion. Easy to see that 6 7 8 9 @ A B C 2 D B E F C . Then 6 G 9 6 H I @ A B P 2 D B E F P Q a. 6 G R 6 G S 9 6 G R T G S . b. 6 G T U V 7 9 6 G 5 c. A line segment from I W X Y Q to I W X ` a Q maps to a circle of radius 6 H centered at origin. d. No Zeros. 3. Trigonometric Functions De• ne @ A B 4 9 6 7 G 2 6 b 7 G ` B E F 4 9 6 7 G c 6 b 7 G ` d e F 4 9 B E F 4 @ A B 4 a. B E F U 4 2 @ A B U 4 9 f b. ` B E F 4 g @ A B 4 U 9 B E F I 4 g 2 4 U Q 2 B E F I 4 g c 4 U Q c. ` @ A B 4 g @ A B 4 U 9 @ A B I 4 g 2 4 U Q 2 @ A B I 4 g c 4 U Q d. ` B E F 4 g B E F 4 U 9 c @ A B I 4 g 2 4 U Q 2 @ A B I 4 g c 4 U Q e. B E F I 4 2 ` a Q 9 B E F 4 and @ A B I 4 2 ` a Q 9 @ A B 4 5 f. B E F 4 9 Y iff 4 9 h a I h 9 Y X i f X 5 5 5 Q g. @ A B 4 9 Y iff 4 9 V U 2 h a I h 9 Y X i f X 5 5 5 Q h. These functions are not bounded. i. A line segment from I Y X P Q to I ` a X P Q maps to an ellipse with semimajor axis equal to @ A B p q under r s t function. 4. Hyperbolic Functions De• ne u v r p w x y €  y ‚ € ƒ r s t p w x y € „ y ‚ € ƒ
Mappings 7 a. … † ‡ ˆ ‰  ‘ ’ “  … † ‡ ‘ ” • – — ˜ ™ d e f g • – — e b. • – — ˜ h e i — j k ˜ h e g l c. — j k ˜ ™ e m n o d f g — j k ˜ ™ e f ” • – — ˜ ™ e m n o d f g • – — ˜ ™ e f d. — j k ˜ e g p iff e g q o d ™ q g p r s l r t t t f e. • – — ˜ e g p iff e g u v h m q o w d ™ q g p r s l r t t t f 5. Logarithmic Function De• ne x y z { | x y z } ~  €  ~ ‚ ƒ „ … then † ‡ ˆ ‰ Š ‹ Œ  a. Is multiple-valued. Hence cannot be considered as inverse of exponetial func- tion. b. Priniciple value of log function is given by Ž   Œ ‹ Ž   ‘ ’ “ ” where ” is the principal value of argument of • . c. – — ˜ ™ • š • › œ  – — ˜ • š ž – — ˜ • › Mappings Ÿ    ™ • œ . Graphical representation of images of sets under   is called ¡ ¢ £ £ ¤ ¥ ¦ § Typi- cally shown in following manner: 1. Draw regular sets (lines, circles, geometric regions etc) in a complex plane, which we call ¨ plane. Use ¨ © ª « ¬ ­ © ® ¯ ° ± ² 2. Show its images on another complex plane, which we call ³ plane. Use ³ © ´ µ ¨ ¶ © · « ¬ ¸ © ¹ º » ¼ . Example 2.3 ½ ¾ ¿ À .
8 Chapter 2 Functions of Complex Variables Mapping of Á  10.80.60.40.20-0.2-0.4-0.6 2 1.5 1 0.5 0 Mapping Á  1. A straight line Ã Ä Å maps to a parabola Æ Ç È É Ê Ë Ì Í Î Ï Ë Ì Ð 2. A straight line Ñ Ò Ë maps to a parabola Ó Ì Ò Ê Ë Ì Í Î Ï Ë Ì Ð 3. A half circle given by Ô Ò Õ Ö × Ø Ù where Ú Û Ü Û Ý maps to a full circle given by Þ Ò Õ Ì Ö × Ø Ì Ù ß This also means that the upper half plane maps on to the entire complex plane. 4. A hyperbola à Ì Ï Ñ Ì Ò á maps to a straight line â ã á ä Mappings by Elementary Functions. 1. Translation by å æ is given by ç ã å è å æ ä 2. Rotation through an angle é æ is given by ç ã ê ë ì å ä 3. Rel• ection through x axis is given by ç ã í î ï 4. Exponential Function
Mappings by Elementary Functions. 9 Exponential Function A vertical line maps to a circle. A horizontal line maps to a radial line. A horizontal strip enclosed between ð ñ ò and ð ñ ó ô maps to the entire complex plane. 5. Sine Function õ ö ÷ ø ñ õ ö ÷ ù ú û õ ü ð ý þ ú û õ ù õ ö ÷ ü ð A vertical line maps to a branch of a hyperbola. A horizontal line maps to an ellipse and has a period of ó ô .
3 Analytic Functions Limits A function ÿ is de• ned in a deleted nbd of   ¡ ¢ De• nition 3.1 The limit of the function ÿ £   ¤ as   ¥ ¦ § is a number ¨ § if, for any given © there exists a such that ! # $ % ' ( ! ) 0 1 Example 3.1 # $ % 2 3 4 5 Show that 6 7 8 9 @ A B C D E F G H E I P Example 3.2 C D E F G E Q P Show that R S T U V U B C D E F G E Q I P Example 3.3 C D E F G E W XE P Show that the limit of C does not exist as E Y ` . Theorem 3.1 Let C D E F G a D b c d F e f g D b c d F and h I G a I e f g I P R S T U V U B C D E F G h I if and only if R S T i p q r s V i p B q r B s a G a I and R S T i p q r s V i p B q r B s g G g I P Example 3.4 C D E F G t S u E . Show that the R S T U V U B C D E F G t S u E I Example 3.5 C D E F G v b e f d Q . Show that the R S T U V Q w C D E F G x f . Theorem 3.2 If R S T U V U B C D E F G h I and R S T U V U B y D E F G €  ‚ ƒ „ … † ‡ † ˆ ‰  ‘ ’ “ ” • ‘ ’ “ – — ˜ ™ d e ™ f g h i j k j l m n o p q n o p — ˜ ™ e ™ f g h i j k j l m n o p r q n o p — ˜ ™ e ™ e ™ s— t u This theorem immediately makes available the entire machinery and tools used for real analysis to be applied to complex analysis. The rules for • nding limits then can be listed as follows:
12 Chapter 3 Analytic Functions 1. v w x y z y { | } | ~ 2. v w x y z y {  € }  €  ~ 3. v w x y z y { ‚ ƒ  „ } ‚ ƒ   „ if ‚ is a polynomial in  . 4. v w x y z y { … † ‡ ƒ  „ } … † ‡ ƒ   „ ~ 5. v w x y z y { ˆ w ‰ ƒ  „ } ˆ w ‰   ~ Continuity De• nition 3.2 A function Š , de•ned in some nbd of   is continuous at   if v w x y z y { Š ƒ  „ } Š ƒ   „ ~ This de• nition clearly assumes that the function is de• ned at   and the limit on the LHS exists. The function Š is continuous in a region if it is continuous at all points in that region. If funtions Š and ‹ are continuous at Œ  then Ž  ‹ , Ž ‹ and Ž  ‹ ‘ ‹ ‘ Œ  ’ “ ” • ’ are also continuous at Œ  . If a function Ž ‘ Œ ’ ” – ‘ — ˜ ™ ’  š › ‘ — ˜ ™ ’ is continuous at Œ  then the component functions – and › are also continuous at ‘ —  ˜ ™  ’ . Derivative De• nition 3.3 A function Ž , de•ned in some nbd of Œ  is differentiable at Œ  if œ  ž Ÿ   Ÿ ¡ Ž ‘ Œ ’ ¢ Ž ‘ Œ  ’ Œ ¢ Œ  exists. The limit is called the derivative of Ž at Œ  and is denoted by Ž £ ¤ ¥ ¦ § or ¨ © ª « ¬ ­ ® ¯ . Example 3.6 ° ¬ ­ ¯ ± ­ ² ³ Show that ° ´ ¬ ­ ¯ ± µ ­ ³ Example 3.7 ° ¬ ­ ¯ ± ¶ ­ ¶ ² . Show that this function is differentiable only at ­ ± · ¸ . In real analysis ¹ º ¹ is not differentiable but ¹ º ¹ » is. If a function is differentiable at ¼ , then it is continuous at ¼ .
Cauchy-Riemann Equations 13 The converse in not true. See Example 3.7. Even if component functions of a complex function have all the partial derivatives, does not imply that the complex function will be differentiable. See Example 3.7. Some rules for obtaining the derivatives of functions are listed here. Let ½ and ¾ be differentiable at ¿ À 1. Á Á Â Ã ½ Ä ¾ Å Ã ¿ Å Æ ½ Ç Ã ¿ Å Ä ¾ Ã ¿ Å À 2. Á Á Â Ã ½ ¾ Å Ã ¿ Å Æ ½ Ç Ã ¿ Å ¾ Ã ¿ Å È ½ Ã ¿ Å ¾ Ç Ã ¿ Å À 3. Á Á Â Ã ½ É ¾ Å Ã ¿ Å Æ Ã ½ Ç Ã ¿ Å ¾ Ã ¿ Å Ê ½ Ã ¿ Å ¾ Ç Ã ¿ Å Å É Ë ¾ Ã ¿ Å Ì Í if ¾ Ã ¿ Å ÎÆ Ï À 4. Á Á Â Ã ½ Ð Ñ Ò Ó Ô Ò Õ Ö × Ó Ñ Ó Ô Ò Ò Ñ × Ó Ô Ò Ø 5. Ù Ù Ú Û Õ Ü Ø 6. Ù Ù Ú Ô Ý Õ Þ Ô Ý ß à Ø Cauchy-Riemann Equations Theorem 3.3 If Ö × Ó Ô á Ò exists, then all the •rst order partial derivatives of component function â Ó ã ä å Ò and æ Ó ã ä å Ò exist and satisfy Cauchy-Riemann Conditions: â ç Õ æ è â è Õ é æ ç Ø Example 3.8 Ö Ó Ô Ò Õ Ô ê Õ ã ê é å ê ë ì í ã å Ø Show that Cauchy-Riemann Condtions are satis•ed. Example 3.9 Ö Ó Ô Ò Õ î Ô î ê Õ ã ê ë å ê Ø Show that the Cauchy-Riemann Condtions are satis•ed only at Ô Õ Ü . Theorem 3.4 Let Ö Ó Ô Ò Õ â Ó ã ä å Ò ë ì æ Ó ã ä å Ò be de•ned in some nbd of the point Ô á Ø If the •rst partial derivatives of â and æ exist and are continuous at Ô á and satisfy Cauchy- Riemann equations at Ô á , then Ö is differentiable at Ô á and Ö × Ó Ô Ò Õ â ç ë ì æ ç Õ æ è é ì â è Ø Example 3.10 Ö Ó Ô Ò Õ ï ð ñ Ó Ô Ò Ø Show that Ö × Ó Ô Ò Õ ï ð ñ Ó Ô Ò Ø Example 3.11 Ö Ó Ô Ò Õ ò ó ô Ó Ô Ò Ø Show that Ö × Ó Ô Ò Õ õ ö ò Ó Ô Ò Ø Example 3.12 Ö Ó Ô Ò Õ ÷ ø ù ø ú Show that the CR conditions are satis•ed at û ü ý but the function þ is not differentiable at ý ú
14 Chapter 3 Analytic Functions If we write ÿ   ¡ ¢ £ ¤ then we can write Cauchy-Riemann Conditions in polar coordi- nates: ¥ ¦   § ¡ ¨ ¤ ¥ ¤   © ¡ ¨ ¦ Analytic Functions De• nition 3.4 A function is analytic in an open set if it has a derivative at each point in that set. De• nition 3.5 A function is analytic at a point ÿ if it is analytic in some nbd of ÿ . De• nition 3.6 A function is an entire function if it is analytic at all points of Example 3.13 ÿ   § ÿ is analytic at all nonzero points. Example 3.14 ÿ   ÿ is not analytic anywhere. A function is not analytic at a point ÿ but is analytic at some point in each nbd of ÿ then ÿ is called the singular point of the function . Harmonic Functions De• nition 3.7 A real valued function ! # $ % is said to be ' ( ) 0 1 2 3 in a domain of xy plane if it has continuous partial derivatives of the •rst and second order and satis•es 4 5 6 7 8 9 @ @ A B C D E F G : H I I P Q R S T U H V P Q R S T W X Y Theorem 3.5 If a function ` P a T W b P Q R S T U c d P Q R S T is analytic in a domain e then the functions f and g are harmonic in e . De• nition 3.8 If two given functions f h i p q r and g h i p q r are harmonic in domain D and their •rst order partial derivatives satisfy Cauchy-Riemann Conditions f s t g u f u t v g s w then g is said to be x y €  ‚ ƒ „ … … ‚ ƒ † ‡ ˆ ‰  ‘ of ’ “ Example 3.15 Let ’ ” • – — ˜ ™ • d e — d and f ” • – — ˜ ™ g • — “ Show that f is hc of ’ and not vice versa. Example 3.16 ’ ” • – — ˜ ™ — h e i • — . Find harmonic conjugate of ’ “
4 Integrals Contours Example 4.1 Represent a line segment joining points j k l m n and o p q r r n by parametric equations. Example 4.2 Show that a half circle in upper half plane with radius s and centered at origin can be parametrized in various ways as given below: 1. t u v w x s y z { v | } u v w x s { ~  v | where v €  ‚ ƒ „ 2. … † ‡ ˆ ‰ ‡ Š ‹ † ‡ ˆ ‰ Œ  Ž  ‡ Ž Š where ‡ €   ‚  „ 3. … † ‡ ˆ ‰  †  ‡  ‘ ˆ Š ‹ † ‡ ˆ ‰   Œ ‡  ‡ Ž Š where ‡ €  ‚ ‘ „ De• nition 4.1 A set of points ’ ‰ † … Š ‹ ˆ in complex plane is called an “ ” • if … ‰ … † ‡ ˆ Š ‹ ‰ ‹ † ‡ ˆ Š † – — ‡ — ˜ ˆ where … † ‡ ˆ and ‹ † ‡ ˆ are continuous functions of the real parameter ‡ „ Example 4.3 … † ‡ ˆ ‰ ™ š › ‡ Š ‹ † ‡ ˆ ‰ › œ  †  ‡ ˆ , where ‡ €  ‚  žƒ . Show that the curve cuts itself and is closed. An arc is called simple if ‡ Ÿ  ‰ ‡ Ž ¡ ’ † ‡ Ÿ ˆ  ‰ ’ † ‡ Ž ˆ „ An arc is closed if ’ † – ˆ ‰ ’ † ˜ ˆ . An arc is differentiable if ’ ¢ † ‡ ˆ ‰ … ¢ † ‡ ˆ £ ¤ ‹ ¢ † ‡ ˆ exists and … ¢ † ‡ ˆ and ‹ ¢ † ‡ ˆ are continu- ous. A smooth arc is differentiable and ’ ¢ † ‡ ˆ is nonzero for all ‡ . De• nition 4.2 Length of a smooth arc is de•ned as ¥ ¦ § ¨ © ª « ¬ ­ ® ¯ ° ± ² ³ ´ µ ² ¶ The length is invariant under parametrization change.
16 Chapter 4 Integrals De• nition 4.3 A · ¸ ¹ º ¸ » ¼ is a constructed by joining •nite smooth curves end to end such that ½ ¾ ¿ À is continuous and ½ Á ¾ ¿ À is piecewise continuous. A closed simple contour has only • rst and last point same and does not cross itself. Contour Integral If  is a contour in complex plane de• ned by ½ ¾ ¿ À Ã Ä ¾ ¿ À Å Æ Ç ¾ ¿ À and a function È ¾ ½ À Ã É ¾ Ä Ê Ç À Å Æ Ë ¾ Ä Ê Ç À is de• ned on it. The integral of È ¾ ½ À along the contour  is denoted and de• ned as follows: Ì Í Î Ï Ð Ñ Ò Ð Ó Ô Õ Ö Î Ï Ð Ñ Ð × Ï Ø Ñ Ò Ø Ó Ô Õ Ö Ï Ù Ú × Û Ü Ý × Ñ Ò Ø Þ ß Ô Õ Ö Ï Ù Ý × Þ Ü Ú × Ñ Ò Ø Ó Ô Ï Ù Ò Ú Û Ü Ò Ý Ñ Þ ß Ô Ï Ù Ò Ý Þ Ü Ò Ú Ñ The component integrals are usual real integrals and are well de• ned. In the last form appropriate limits must placed in the integrals. Some very straightforward rules of integration are given below: 1. à á â ã ä å æ ç å è â à á ã ä å æ ç å where â is a complex constant. 2. à á ä ã ä å æ é ê ä å æ æ ç å è à á ã ä å æ ç å é à á ê ä å æ ç å ë 3. à á ì í á î ã ä å æ ç å è à á ì ã ä å æ ç å é à á î ã ä å æ ç å ë 4. ï ï ð ñ ò ó ô õ ö ô ï ï ÷ ð ñ ø ò ó ô ó ù õ õ ô ú ó ù õ ø ö ù û 5. If ø ò ó ô õ ø ÷ ü for all ý þ ÿ then     ¡ ¢ £ ¤ ¥ ¦ § ¥     ¨ © , where is length of the countour ÿ Example 4.4 £ ¤ ¥ ¦ ¥ Find integral of £ from ¤ ¦ to ¤ ¦ along a straight line and also along st line path from ¤ ¦ to ¤ ¦ and from ¤ ¦ to ¤ ¦ . Example 4.5 £ ¤ ¥ ¦ ¥ . Find the integral from ¤ ¦ to ¤ ¦ along a semicircu- lar path in upper plane given by ¥ Example 4.6 Show that     ¡ ¢ £ ¤ ¥ ¦ § ¥     ¨ ! for £ ¤ ¥ ¦ # ¥ $ and ÿ % ¥ from to
Cauchy-Goursat Theorem 17 Cauchy-Goursat Theorem Theorem 4.1 (Jordan Curve Theorem) Every simple and closed contour in complex plane splits the entire plane into two domains one of which is bounded. The bounded domain is called the interior of the countour and the other one is called the exterior of the contour. De• ne a sense direction for a contour. Theorem 4.2 Let ' be a simple closed contour with positive orientation and let ( be the interior of ' ) If 0 and 1 are continuous and have continuous partial derivatives 2 3 4 2 5 4 1 3 and 1 5 at all points on 6 and 7 , then 8 9 @ 2 @ A 4 B C D A E 1 @ A 4 B C D B C F 8 8 G 1 3 @ A 4 B C H 2 5 @ A 4 B C I D A D B Theorem 4.3 (Cauchy-Goursat Theorem) Let P be analytic in a simply connected domain 7 Q If 6 is any simple closed contour in 7 , then 8 9 P @ R C D R F S Q Example 4.7 P @ R C F R T 4 U V W @ R C 4 X Y ` @ R C etc are entire functions so integral about any loop is zero. Theorem 4.4 Let 6 a and 6 T be two simple closed positively oriented contours such that 6 T lies entirely in the interior of 6 a Q If P is an analytic function in a domain 7 that contains 6 a and 6 T both and the region between them, then 8 9 b P @ R C D R F 8 9 c P @ R C D R Q Example 4.8 P @ R C F d e R . Find f 9 P @ R C D R if 6 is any contour containing origin. Choose a circular contour inside 6 Q Example 4.9 f 9 a g h g i D R F p q r if 6 contains R s Q Example 4.10 Find f 9 T g t g g c u T where 6 v w R w F p Q Extend the Cauchy Goursat theorem to multiply connected domains. Antiderivative Theorem 4.5 (Fundamental Theorem of Integration) Let P be de•ned in a simply connected domain 7 and is analytic in 7 . If R s and R are points in 7 and 6 is any contour in 7 joining R s and R 4 then the function x @ R C F 8 9 P @ R C D R
18 Chapter 4 Integrals is analytic in y and €  ‚ ƒ „ … † ‚ ƒ „ ‡ De• nition 4.4 If † is analytic in y and ƒ ˆ and ƒ ‰ are two points in y then the de•nite integral is de•ned as  ‘ ’  “ † ‚ ƒ „ ” ƒ … € ‚ ƒ ‰ „ • € ‚ ƒ ˆ „ where € is an antiderivative of † . Example 4.11 ‰ – — ˜ ™ ƒ ‰ ” ƒ … ƒ d e f g g ‰ – — ™ … ‰ d h i ˆ ˆ d ‡ Example 4.12 — ˜ ˆ j k l ƒ … l m n ƒ o — ˆ … l m n i • l m n p ‡ Example 4.13  ‘ ˜  “ q   … r k s ƒ ‰ • r k s ƒ ˆ ‡ Cauchy Integral Formula Theorem 4.6 (Cauchy Integral Formula) Let † be analytic in domain y . Let t be a positively oriented simple closed contour in y . If ƒ ™ is in the interior of t then † ‚ ƒ ™ „ … p u v i ’ w † ‚ ƒ „ ” ƒ ƒ • ƒ ™ ‡ Example 4.14 † ‚ ƒ „ … ˆ  ‘ – x ‡ Find ˜ w † ‚ ƒ „ ” ƒ if t y o ƒ • i o … u ‡ Example 4.15 † ‚ ƒ „ …  ‰  – ˆ ‡ Find ˜ w † ‚ ƒ „ ” ƒ if t is square with vertices on ‚ z u { z u „ ‡ Theorem 4.7 If † is analytic at a point, then all its derivatives exist and are analytic at that point. † | } ~ ‚ ƒ ™ „ …  € u v i ’ w † ‚ ƒ „ ” ƒ ‚ ƒ • ƒ ™ „ } – ˆ
5 Series Convergence of Sequences and Series Example 5.1  ‚ ƒ „ … † Example 5.2  ‚ ƒ ‡ ‚ Example 5.3  ‚ ƒ ˆ ‰ Š ‹ Œ ˆ „  ‚ … † Ž De• nition 5.1 An in•nite sequece     Ž  ‘ ‘ ‘   ‚  ‘ ‘ ‘ of complex numbers has a ’ “ ” “ •  if for each positive – , there exists positive integer — such that ˜ ™ š › ™ ˜ œ  ž Ÿ   ¡   ¢ £ ¤ ¥ ¦ § ¨ The sequences have only one limit. A sequence said to converge to © if © is its limit. A sequence diverges if it does not converge. Example 5.4 © ª « © ª converges to 0 if ¬ © ¬ ­ ® else diverges. Example 5.5 © ª « ® ¯ ° ¥ ± ² ³ ¥ ± ® ´ ¯ ¥ converges to ² . Theorem 5.1 Suppose that © ª « µ ª ± ² ¶ ª and © « µ ± ² ¶ ¨ Then, · ¸ ¹ ª º » ¼ ½ ¾ ¼ if and only if ¿ À Á ½  » à ½ ¾ Ã Ä Å Æ Ç È É Ê Ë Ì Í Ê Î Í De• nition 5.2 If Ï Ð Ê Ñ is a sequence, the in•nite sum Ð Ò Ó Ð Ô Ó Õ Õ Õ Ó Ð Ê Ó Õ Õ Õ is called a series and is denoted by Ì Ö × Ø Ù Ú Û . De• nition 5.3 A series Ü Ý Û Ø Ù Ú Û is said to converge to a sum Þ if a sequence of partial sums Þ ß à á â ã á ä ã å å å ã á æ converges to ç .
20 Chapter 5 Series Theorem 5.2 Suppose that è é ê ë é ì í î é and ï ê ð ñ ò ó ô Then, õ ö ÷ ø ù ú ÷ û ü if and only if ý ö ÷ ø ù þ ÷ û ÿ   ¡ ¢ ý ö ÷ ø ù £ ÷ û ¤ Example 5.6 ¥ ý ¦ ú ÷ û § ¨ © § ú if ú § Taylor Series Theorem 5.3 (Taylor Series) If is analytic in a circular disc of radius ¦ andcentered at ú ¦ then at each point inside the disc there is a series representation for given by © ú û ý ö ÷ ø ¦ ÷ © ú ú ¦ ÷ where ÷ û ÷ © ú ¦ ! Example 5.7 # ¡ ú û ¥ ý ¦ ú $ ÷ % ù ¨ © ' § ! ( Example 5.8 ù ù % ) û ¥ ý ¦ © § ÷ ú ÷ ú § ( Example 5.9 0 ) û ¥ ý ¦ ú ÷ ¨ ! ( Example 5.10 ù ) û ¥ ý ¦ © § ÷ © ú § ÷ ú § § ( Laurent Series Theorem 5.4 (Laurent Series) If is analytic at all points ú in an annular region 1 such that ù ú ú ¦ $ then at each point in 1 there is a series representation for given by © ú û ý ö ÷ ø ¦ ÷ © ú ú ¦ ÷ ' ý ö ÷ ø ù 2 ÷ © ú ú ¦ ÷ where ÷ û § 3 4 5 6 © ú 7 ú © ú ú ¦ ÷ % ù 2 ÷ û § 3 4 5 6 © ú 7 ú © ú ú ¦ 8 ÷ % ù
Laurent Series 21 and 9 is any contour in @ . Example 5.11 If A is analytic inside a disc of radius B about C D E then the Laurent series for A is identical to the Taylor series for A . That is all F G H I . Example 5.12 P P Q R H S T G U P V W P X Y ` a b c where d e d f g h Example 5.13 i p e q r s t u b s t v u b s w v h Find Laurent series for all d e d x g y g x d e d x € and d e d f € h Example 5.14 Note that  t r t w ‚ ƒ „ … i p e q † e h
6 Theory of Residues And Its Applications Singularities De• nition 6.1 If a function ‡ fails to be analytic at ˆ ‰ but is analytic at some point in each neighbourhood of ˆ ‰ , then ˆ ‰ is a  ‘ ’ “ ” • – — ˜ ™ ‘ ’ d of e . De• nition 6.2 If a function e fails to be analyitc at f g but is analytic at each f in h i j f k f g j i l for some l m then e is said to be an ‘  ™ • – d n o p q r s t u v w x y q r z of { . Example 6.1 { | } ~  €  } has an isolated singularity at ‚ . Example 6.2 { | } ~  €  ƒ „ … | † } ~ has isolated singularities at }  ‚ ‡ ˆ € ‡ ‰ ‰ ‰ . Example 6.3 { | } ~  €  ƒ „ … | †  } ~ has isolated singularities at }  €  Š for integral Š , also has a singularity at }  ‚ . Example 6.4 { | } ~  ‹ Œ  } all points of negative x-axis are singular. Types of singularities If a function { has an isolated singularity at } Ž then  a  such that ‘ is analytic at all points in ’ “ ” • – • — ” “  . Then ‘ must have a Laurent series expansion about • — . The part ˜ ™š › œ  œ ž • – • — Ÿ   š is called the principal part of ‘ ¡ 1. If there are in• nite nonzero  ¢ in the prinipal part then • — is called an essential singu- larity of ‘ ¡ 2. If for some integer £ ¤ ¥ ¦ § ¨ © but ª « ¨ © for all ¬ ­ ® then ¯ ° is called a pole of order ® of ± ² If ® ¨ ³ then it is called a simple pole. 3. If all ª s are zero then ¯ ° is called a removable singularity.
24 Chapter 6 Theory of Residues And Its Applications Example 6.5 ´ µ ¶ · ¸ µ ¹ º » ¶ · ¼ ¶ is unde•ned at ¶ ¸ ½ ¾ ´ has a removable isolated singularity at ¶ ¸ ½ ¾ Residues Suppose a function ´ has an isolated singularity at ¶ ¿ À then there exists a Á Â ½ such that ´ is analytic for all ¶ in deleted nbd ½ Ã Ä ¶ Å ¶ ¿ Ä Ã Á . Then ´ has a Laurent series representation ´ µ ¶ · ¸ Æ Ç È É ¿ Ê È µ ¶ Å ¶ ¿ · È Ë Æ Ç È É Ì Í È µ ¶ Å ¶ ¿ · È ¾ The coef• cient Í Ì ¸ Î Ï Ð Ñ Ò Ó ´ µ ¶ · Ô ¶ where Õ is any contour in the deleted nbd, is called the residue of ´ at ¶ ¿ ¾ Example 6.6 ´ µ ¶ · ¸ Ì Ö × Ö Ø . Then Ù Ó ´ µ ¶ · Ô ¶ ¸ Ï Ð Ñ Í Ì ¸ Ï Ð Ñ if Õ contains ¶ ¿ , other- wise ½ . Example 6.7 ´ µ ¶ · ¸ Ì Ö Ú Ö × Ö Ø Û Ü Ý Show Þ ß à á â ã ä â å æ ç è é if ê ë ì í î ï ì ð ñ ò Example 6.8 ó ô í õ ð í ö ÷ ø ù ú û ü . Show ý þ ó ô í õ ÿ í ð   ¡ if ê ë ì í ì ð ñ ò Example 6.9 ó ô í õ ð ö ÷ ø ù ú û ¢ ü . Show ý þ ó ô í õ ÿ í ð £ if ê ë ì í ì ð ñ even though it has a singularity at í ð £ ò Theorem 6.1 If a function ó is analytic on and inside a positively oriented countour ê , except for a •nite number of points í ú ¤ í ¥ ¦ ò ò ò ¦ í § inside ¨ , then © § ! # Res ! $ Example 6.10 Show that % ' ( ) ' 0 ' ( 1 2 3 4 5 6 7 8 9 Residues of Poles Theorem 6.2 If a function @ has a pole of order A at 4 B then Res@ C 4 B D 5 E F G ' H ' I P Q R S T Q U V W X Y ` W a X Y ` R R a T a b U X c R a U U d e Example 6.11 f g h i p q r s r t e Simple pole Res f g h u i p v e
Quotients of Analytic Functions 25 Example 6.12 w x y €  ‚ ƒ „ ƒ … ƒ † ‡ ˆ . Simple pole at y  ‰  Res w x ‰ €  ‘ ’ ‘ “ . Pole of order ” at y  • . Res w x • €  – ‘ ’ ‘ “ . Example 6.13 w x y €  — ˜ ™ d d e f d g h i j . Res k l m n o p q r s t . Res k l s n o p u v p s w x r y s t z Quotients of Analytic Functions Theorem 6.3 If a function k l { n o | } ~  €  ‚ ƒ , where „ and … are analytic at † ‡ ˆ then 1. ‰ is singular at † ‡ iff … Š † ‡ ‹ Œ  Ž 2. ‰ has a simple pole at † ‡ if …  Š † ‡ ‹ Œ  . Then residue of ‰ at † ‡ is „ Š † ‡ ‹ ‘ …  Š † ‡ ‹ .
A References This appendix contains the references.
B Index This appendix contains the index.

More Related Content

PPT
27 triple integrals in spherical and cylindrical coordinates
bymath267
 
PDF
Complex function
byDr. Nirav Vyas
 
PPTX
Complex analysis
bysujathavvv
 
PDF
Multiple integrals
byTarun Gehlot
 
PDF
Indefinite and Definite Integrals Using the Substitution Method
byScott Bailey
 
PPTX
Series solutions at ordinary point and regular singular point
byvaibhav tailor
 
PPTX
Unit 1: Topological spaces (its definition and definition of open sets)
bynasserfuzt
 
PPTX
Matrix of linear transformation
bybeenishbeenish
 
27 triple integrals in spherical and cylindrical coordinates
bymath267
 
Complex function
byDr. Nirav Vyas
 
Complex analysis
bysujathavvv
 
Multiple integrals
byTarun Gehlot
 
Indefinite and Definite Integrals Using the Substitution Method
byScott Bailey
 
Series solutions at ordinary point and regular singular point
byvaibhav tailor
 
Unit 1: Topological spaces (its definition and definition of open sets)
bynasserfuzt
 
Matrix of linear transformation
bybeenishbeenish
 

What's hot

PDF
Ordinary differential equations
bySusana I. F.
 
PPTX
Graph Theory
bykailash shaw
 
PPT
Asymptotes and holes 97
byswartzje
 
PDF
Eigenvalue eigenvector slides
byAmanSaeed11
 
PPTX
Complex integration
bySanthanam Krishnan
 
PPSX
Complex Analysis - Differentiability and Analyticity (Team 2) - University of...
byAlex Bell
 
PPT
Applications of Integrations
byitutor
 
PPTX
Laplace transformation
byamanullahkakar2
 
PPT
Integration Ppt
byakhilkhanna
 
PPTX
Cramer row inequality
byVashuGupta8
 
PPTX
Curves in space
byPiyaliDey14
 
PPTX
Matrix of linear transformation 1.9-dfs
byFarhana Shaheen
 
PPT
24 double integral over polar coordinate
bymath267
 
PDF
Quadratic programming (Tool of optimization)
byVARUN KUMAR
 
PPT
Partial-Orderings in Discrete Mathematics
byMeghaj Mallick
 
PDF
Chi square distribution
byA. Dally Maria Evangeline
 
PPT
Lesson 14 a - parametric equations
byJean Leano
 
PPT
Power series convergence ,taylor & laurent's theorem
byPARIKH HARSHIL
 
PPT
inverse z-transform ppt
bymihir jain
 
PPTX
Vector Spaces,subspaces,Span,Basis
byRavi Gelani
 
Ordinary differential equations
bySusana I. F.
 
Graph Theory
bykailash shaw
 
Asymptotes and holes 97
byswartzje
 
Eigenvalue eigenvector slides
byAmanSaeed11
 
Complex integration
bySanthanam Krishnan
 
Complex Analysis - Differentiability and Analyticity (Team 2) - University of...
byAlex Bell
 
Applications of Integrations
byitutor
 
Laplace transformation
byamanullahkakar2
 
Integration Ppt
byakhilkhanna
 
Cramer row inequality
byVashuGupta8
 
Curves in space
byPiyaliDey14
 
Matrix of linear transformation 1.9-dfs
byFarhana Shaheen
 
24 double integral over polar coordinate
bymath267
 
Quadratic programming (Tool of optimization)
byVARUN KUMAR
 
Partial-Orderings in Discrete Mathematics
byMeghaj Mallick
 
Chi square distribution
byA. Dally Maria Evangeline
 
Lesson 14 a - parametric equations
byJean Leano
 
Power series convergence ,taylor & laurent's theorem
byPARIKH HARSHIL
 
inverse z-transform ppt
bymihir jain
 
Vector Spaces,subspaces,Span,Basis
byRavi Gelani
 

Viewers also liked

PPT
Mathematics and History of Complex Variables
bySolo Hermelin
 
PPT
Prime numbers
bySolo Hermelin
 
PPT
Complex varible
byNaveen Sihag
 
PPT
Vector analysis
bySolo Hermelin
 
PPT
Introduction to Mathematical Probability
bySolo Hermelin
 
PPT
Matrices i
bySolo Hermelin
 
PPSX
Lecture notes for s4 b tech Mathematics
byAnoop T Vilakkuvettom
 
PPTX
Matrix Groups and Symmetry
byFranklin College Mathematics and Computing Department
 
PDF
Golden words of swami vivekananda
bychiefs world
 
PDF
Swami vivekananda’s 150 quotes
byFiroj Md Shah
 
PPT
Swami Vivekananda Quotes
byMoncy Varghese
 
PPTX
Integrating spreadsheets, visualization tools, and computational knowledge en...
byFranklin College Mathematics and Computing Department
 
Mathematics and History of Complex Variables
bySolo Hermelin
 
Prime numbers
bySolo Hermelin
 
Complex varible
byNaveen Sihag
 
Vector analysis
bySolo Hermelin
 
Introduction to Mathematical Probability
bySolo Hermelin
 
Matrices i
bySolo Hermelin
 
Lecture notes for s4 b tech Mathematics
byAnoop T Vilakkuvettom
 
Matrix Groups and Symmetry
byFranklin College Mathematics and Computing Department
 
Golden words of swami vivekananda
bychiefs world
 
Swami vivekananda’s 150 quotes
byFiroj Md Shah
 
Swami Vivekananda Quotes
byMoncy Varghese
 
Integrating spreadsheets, visualization tools, and computational knowledge en...
byFranklin College Mathematics and Computing Department
 

Similar to Complex analysis book by iit

PDF
Complex Numbers and Functions. Complex Differentiation
byHesham Ali
 
PPT
Complex Variables SOLO HERMELIN 222222222222222222222222
bylinodi9618
 
PDF
Hand book of Howard Anton calculus exercises 8th edition
byPriSim
 
PDF
Santos, david precalculus -notes
byKarina Ruiz
 
PDF
Maths short notes for JEE.pdfvgvggvggygyvtgtb
bymohithkeerthan9
 
PDF
Maths short dhhdbrbdhd hdh hdb rh notes for JEE.pdf
bybagariakartik123
 
PDF
Nbhm m. a. and m.sc. scholarship test 2008
byMD Kutubuddin Sardar
 
PDF
01. Functions-Theory & Solved Examples Module-4.pdf
byRajuSingh806014
 
PDF
A First Course In Complex Analysis
byNathan Mathis
 
PDF
2ndyear20yrs
byMUNEEB KHAN
 
PDF
Chapter 1 (math 1)
byAmr Mohamed
 
PDF
Complex
byAndrei Petrescu
 
PDF
Ahlfors sol1
byMonu Jangid
 
PDF
Top schools in delhi ncr
byEdhole.com
 
PPT
453015708-FUNGSI-EKSPONENSIAL-2020-ppt.ppt
bynila87
 
PDF
Function
byQamarAbbasJaffari
 
PDF
chap 2 Ex#1.1
byAns Ali
 
PDF
multivariate-CalculusCHAP1_Complex_Analysis.pdf
byMaaz609108
 
PDF
Lecture 1
bySaeedShahAhmed
 
PDF
Math17 Reference
byElise Angela Espinosa
 
Complex Numbers and Functions. Complex Differentiation
byHesham Ali
 
Complex Variables SOLO HERMELIN 222222222222222222222222
bylinodi9618
 
Hand book of Howard Anton calculus exercises 8th edition
byPriSim
 
Santos, david precalculus -notes
byKarina Ruiz
 
Maths short notes for JEE.pdfvgvggvggygyvtgtb
bymohithkeerthan9
 
Maths short dhhdbrbdhd hdh hdb rh notes for JEE.pdf
bybagariakartik123
 
Nbhm m. a. and m.sc. scholarship test 2008
byMD Kutubuddin Sardar
 
01. Functions-Theory & Solved Examples Module-4.pdf
byRajuSingh806014
 
A First Course In Complex Analysis
byNathan Mathis
 
2ndyear20yrs
byMUNEEB KHAN
 
Chapter 1 (math 1)
byAmr Mohamed
 
Complex
byAndrei Petrescu
 
Ahlfors sol1
byMonu Jangid
 
Top schools in delhi ncr
byEdhole.com
 
453015708-FUNGSI-EKSPONENSIAL-2020-ppt.ppt
bynila87
 
Function
byQamarAbbasJaffari
 
chap 2 Ex#1.1
byAns Ali
 
multivariate-CalculusCHAP1_Complex_Analysis.pdf
byMaaz609108
 
Lecture 1
bySaeedShahAhmed
 
Math17 Reference
byElise Angela Espinosa
 

Recently uploaded

PDF
222109_MD_MILTON_HOSSAIN_26TH_BATCH_REGULAR_PMIT V5.pdf
byMILTONKHAN5
 
PDF
Introduction to MySQL Spatial Features and Real-World Use Cases
byVissarion Fisikopoulos
 
PPTX
K-nearest neighbouring machine learing algorithm .pptx
bydeepalishinkar1
 
PDF
November_2025 Top 10 Read Articles in Computer Networks & Communications.pdf
byijcncjournal019
 
PDF
@Regenerative braking system of DC motor
bysangamkumar993613
 
PPTX
Supercapacitor.pptx...............................
byTarikBouchala
 
PPTX
TRANSPORTATION ENGINEERING Unit-5.1.pptx
byMood Naik
 
PPTX
Generative AI Deep Dive: Architectures, Mechanics, and Future Applications
bygenz9514
 
PPTX
Mc25104 - data structures and algorithms using PYTHON OOP_Python_Lecture_Note...
byjanaki raman
 
PPT
399-Cathodic-Protection-Presentation.ppt
byMekine
 
PPTX
Computer engineering for collage studen. pptx
byestradageric
 
PPTX
2-Photoelectric effect, phenomena and its related concept.pptx
byadhidwih
 
PDF
Reinforced Earth Walls Notes .pdf
byPrasad Prabhu
 
PPTX
Blockchain and cryptography Lecture Notes
bysanjanakumar34
 
PDF
Welcome to ISPR 2026 - 12th International Conference on Image and Signal Pro...
byieijjournal
 
PPTX
Ship Repair and fault diagnosis and restoration of system back to normal .pptx
byIntuition1
 
PPTX
31.03.24 - 7.CURRICULUM & TEACHING - LEARNING PROCESS IMPLEMENTATION DETAILS....
byssuserb3d3fc
 
PPTX
waste to energy deck v.3.pptx changing garbage to electricity
byGegaGrandeza1
 
PPTX
Waste to Energy - G2 Ethanol.pptx to process
byGegaGrandeza1
 
PPTX
Presentation 1.pptx WHAT IS ARTIFICIAL INTELLIGENCE?
bypraveenbubnale77
 
222109_MD_MILTON_HOSSAIN_26TH_BATCH_REGULAR_PMIT V5.pdf
byMILTONKHAN5
 
Introduction to MySQL Spatial Features and Real-World Use Cases
byVissarion Fisikopoulos
 
K-nearest neighbouring machine learing algorithm .pptx
bydeepalishinkar1
 
November_2025 Top 10 Read Articles in Computer Networks & Communications.pdf
byijcncjournal019
 
@Regenerative braking system of DC motor
bysangamkumar993613
 
Supercapacitor.pptx...............................
byTarikBouchala
 
TRANSPORTATION ENGINEERING Unit-5.1.pptx
byMood Naik
 
Generative AI Deep Dive: Architectures, Mechanics, and Future Applications
bygenz9514
 
Mc25104 - data structures and algorithms using PYTHON OOP_Python_Lecture_Note...
byjanaki raman
 
399-Cathodic-Protection-Presentation.ppt
byMekine
 
Computer engineering for collage studen. pptx
byestradageric
 
2-Photoelectric effect, phenomena and its related concept.pptx
byadhidwih
 
Reinforced Earth Walls Notes .pdf
byPrasad Prabhu
 
Blockchain and cryptography Lecture Notes
bysanjanakumar34
 
Welcome to ISPR 2026 - 12th International Conference on Image and Signal Pro...
byieijjournal
 
Ship Repair and fault diagnosis and restoration of system back to normal .pptx
byIntuition1
 
31.03.24 - 7.CURRICULUM & TEACHING - LEARNING PROCESS IMPLEMENTATION DETAILS....
byssuserb3d3fc
 
waste to energy deck v.3.pptx changing garbage to electricity
byGegaGrandeza1
 
Waste to Energy - G2 Ethanol.pptx to process
byGegaGrandeza1
 
Presentation 1.pptx WHAT IS ARTIFICIAL INTELLIGENCE?
bypraveenbubnale77
 

Complex analysis book by iit

  • 1.
    Complex Analysis PH 503CourseTM Charudatt Kadolkar Indian Institute of Technology, Guwahati
  • 2.
    ii Copyright   ¡ 2000 byCharudatt Kadolkar
  • 3.
    Preface Preface Head These noteswere prepared during the lectures given to MSc students at IIT Guwahati, July 2000 and 2001.. Acknowledgments As of now none but myself IIT Guwahati Charudatt Kadolkar.
  • 4.
    Contents Preface iii Preface Headiii Acknowledgments iii 1 Complex Numbers 1 De• nitions 1 Algebraic Properties 1 Polar Coordinates and Euler Formula 2 Roots of Complex Numbers 3 Regions in Complex Plane 3 2 Functions of Complex Variables 5 Functions of a Complex Variable 5 Elementary Functions 5 Mappings 7 Mappings by Elementary Functions. 8 3 Analytic Functions 11 Limits 11 Continuity 12 Derivative 12 Cauchy-Riemann Equations 13
  • 5.
    vi Contents Analytic Functions14 Harmonic Functions 14 4 Integrals 15 Contours 15 Contour Integral 16 Cauchy-Goursat Theorem 17 Antiderivative 17 Cauchy Integral Formula 18 5 Series 19 Convergence of Sequences and Series 19 Taylor Series 20 Laurent Series 20 6 Theory of Residues And Its Applications 23 Singularities 23 Types of singularities 23 Residues 24 Residues of Poles 24 Quotients of Analytic Functions 25 A References 27 B Index 29
  • 6.
    1 Complex Numbers De•nitions De• nition 1.1 Complex numbers are de•ned as ordered pairs ¢ £ ¤ ¥ ¦ Points on a complex plane. Real axis, imaginary axis, purely imaginary numbers. Real and imaginary parts of complex number. Equality of two complex numbers. De• nition 1.2 The sum and product of two complex numbers are de•ned as follows: § ¨ © © ! # $ % ' # $ ' % ( ! # ' # ( ) 0 1 2 3 4 2 5 6 0 1 7 1 2 8 9 @ 9 A B C @ 9 A D C A 9 @ E In the rest of the chapter use F G F H G F I G P P P for complex numbers and Q R S for real numbers. introduce T and U V W X T Y notation. Algebraic Properties 1. Commutativity U ` X U a V U a X U ` b U ` U a V U a U ` 2. Associativity c U ` X U a d X U e f g h i p g q i g e r s p g h g q r g e f g h p g q g e r 3. Distributive Law g p g h i g q r f g g h i g g q 4. Additive and Multiplicative Indentity g i t u v w v x y €  5. Additive and Multiplicative Inverse ‚  € ƒ ‚ „ … ‚ † ‡  ˆ ‰  ‘ ’ “ ” • – ” — ˜ – “ ” • – ” ™ d e f g h
  • 7.
    2 Chapter 1Complex Numbers 6. Subtraction and Division i j k i l m i j n o k i l p q i j i l m i j i r j l 7. Modulus or Absolute Value s t s u v w x y z x 8. Conjugates and properties { | w } ~ z |  w € } z  { ‚ ƒ „ … † „ ‡ ƒ „ … „ ‡ „ … † „ ‡ „ … ˆ „ ‡ „ … ‰ † „ ‡ „ … 9. Š „ Š … † „ „ ‹ Œ  Ž     ‘ ’ “ ” • ” – ” — ˜ 10. Triangle Inequality ™ ” š › ” œ ™  ž Ÿ   ž ¡ ž Ÿ ¢ ž Polar Coordinates and Euler Formula 1. Polar Form: for Ÿ £ ¤ ¥ , Ÿ ¤ ¦ § ¨ © ª « ¬ ­ ® ¯ ° ± ² where ³ ´ µ ¶ µ and · ¸ ¹ º » ¼ ½ ¾ . ¿ is called the argument of À . Since ¿ Á Â Ã Ä is also an argument of Å Æ the principle value of argument of Å is take such that Ç Ä È É Ê Ë Ì For Í Î Ï the Ð Ñ Ò Ó is unde• ned. 2. Euler formula: Symbollically, Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ Û ß à Ü á 3. â ã â ä × å ã å ä æ ç è é ê ë ì í î ï ð ï ñ ò ó ð ó ñ ô õ ö ÷ ø ù ÷ ú î ï û ü ý û þ û ÿ 4. de Moivre’s Formula   ¡ ¢ £ ¤ ¥ ¦ £ § ¨ ¤ © û ü ¡ ¢ £ ¤ ¥ ¦ £ § ¨ ¤
  • 8.
    Roots of ComplexNumbers 3 Roots of Complex Numbers Let then ! # $ % ' ( ) 0 1 2 34 5 5 There are only 4 distinct roots which can be given by 2 6 7 8 9 8 @ @ @ 8 4 A 9 @ If B is a principle value of C D E F then B G 4 is called the principle root. Example 1.1 The three possible roots of H I P Q R S T U V W X Y ` a b c d e f g h are i p q g f r s i p q g f r t p r q g h s i p q g f r t p u q g h . Regions in Complex Plane 1. v w x y € of  ‚ is de• ned as a set of all points ƒ which satisfy „ ƒ … ƒ ‚ „ † ‡ 2. ˆ ‰  ‘ ’ “ ” • – — of ˜ ™ is a nbd of ˜ ™ excluding point ˜ ™ . 3. Interior Point, Exterior Point, Boundary Point, Open set and closed set. 4. Domain, Region, Bounded sets, Limit Points.
  • 9.
    2 Functions ofComplex Vari- ables Functions of a Complex Variable A d e f g h i j k l de• ned on a set m is a rule that uniquely associates to each point n of m a a complex number o p Set q is called the r s t u v w of x and y is called the value of x at z and is denoted by x { z | } y ~ x { z | } x {  €  ‚ | } ƒ „ … † ‡ ˆ ‰ Š ‹ Œ  Ž   ‘ Œ ’  “ ‘ Œ ” • – —  “ ˜ Œ ” Ž ™  š › ‹ Œ ” Ž ™  “ œ  ž Ÿ   ¡ ¢ £ ¤ ¥ ¦ § ¨ © Example 2.1 Write ª « ¬ ­ ® ¯ ° ¬ ± in ² ³ ´ µ form. ² « ¶ · ¸ ­ ® ¹ º » ¼ ½ ¾ ¿ ½ À ¼ ½ Á ½ and Â Ã Ä Å Æ Ç È É ¿ ¼ ¾ ¿ ½ À ¼ ½ Á ½ Ê Ã Ë Å Ì Ç È Ë Í É Î Ï Ð Ñ Ì and Â Ã Ë Å Ì Ç È Ò Ë Í É Ð Ó Ô Ñ Ì Domain of Õ is Ö × Ø Ù Ú . A Û Ü Ý Þ ß à á â ã ä å æ ç è é ê ç ë ì í î ï ë is a rule that assigns more than one value to each point of domain. Example 2.2 ð ñ ò ó ô õ ö ÷ This function assigns two distinct values to each ö ø ù ú û ü . One can choose the function to be single-valued by specifying õ ö ú ý õ þ ÿ   ¡ ¢ £ where ¤ is the principal value. Elementary Functions
  • 10.
    6 Chapter 2Functions of Complex Variables 1. Polynomials ¥ ¦ § ¨ © whrere the coef• cients are real. Rational Functions. 2. Exponential Function ! # # $ % ' ( ) 0 1 2 3 3 3 Converges for all 4 5 For real 4 the de• nition coincides with usual exponential func- tion. Easy to see that 6 7 8 9 @ A B C 2 D B E F C . Then 6 G 9 6 H I @ A B P 2 D B E F P Q a. 6 G R 6 G S 9 6 G R T G S . b. 6 G T U V 7 9 6 G 5 c. A line segment from I W X Y Q to I W X ` a Q maps to a circle of radius 6 H centered at origin. d. No Zeros. 3. Trigonometric Functions De• ne @ A B 4 9 6 7 G 2 6 b 7 G ` B E F 4 9 6 7 G c 6 b 7 G ` d e F 4 9 B E F 4 @ A B 4 a. B E F U 4 2 @ A B U 4 9 f b. ` B E F 4 g @ A B 4 U 9 B E F I 4 g 2 4 U Q 2 B E F I 4 g c 4 U Q c. ` @ A B 4 g @ A B 4 U 9 @ A B I 4 g 2 4 U Q 2 @ A B I 4 g c 4 U Q d. ` B E F 4 g B E F 4 U 9 c @ A B I 4 g 2 4 U Q 2 @ A B I 4 g c 4 U Q e. B E F I 4 2 ` a Q 9 B E F 4 and @ A B I 4 2 ` a Q 9 @ A B 4 5 f. B E F 4 9 Y iff 4 9 h a I h 9 Y X i f X 5 5 5 Q g. @ A B 4 9 Y iff 4 9 V U 2 h a I h 9 Y X i f X 5 5 5 Q h. These functions are not bounded. i. A line segment from I Y X P Q to I ` a X P Q maps to an ellipse with semimajor axis equal to @ A B p q under r s t function. 4. Hyperbolic Functions De• ne u v r p w x y €  y ‚ € ƒ r s t p w x y € „ y ‚ € ƒ
  • 11.
    Mappings 7 a. …† ‡ ˆ ‰  ‘ ’ “  … † ‡ ‘ ” • – — ˜ ™ d e f g • – — e b. • – — ˜ h e i — j k ˜ h e g l c. — j k ˜ ™ e m n o d f g — j k ˜ ™ e f ” • – — ˜ ™ e m n o d f g • – — ˜ ™ e f d. — j k ˜ e g p iff e g q o d ™ q g p r s l r t t t f e. • – — ˜ e g p iff e g u v h m q o w d ™ q g p r s l r t t t f 5. Logarithmic Function De• ne x y z { | x y z } ~  €  ~ ‚ ƒ „ … then † ‡ ˆ ‰ Š ‹ Œ  a. Is multiple-valued. Hence cannot be considered as inverse of exponetial func- tion. b. Priniciple value of log function is given by Ž   Œ ‹ Ž   ‘ ’ “ ” where ” is the principal value of argument of • . c. – — ˜ ™ • š • › œ  – — ˜ • š ž – — ˜ • › Mappings Ÿ    ™ • œ . Graphical representation of images of sets under   is called ¡ ¢ £ £ ¤ ¥ ¦ § Typi- cally shown in following manner: 1. Draw regular sets (lines, circles, geometric regions etc) in a complex plane, which we call ¨ plane. Use ¨ © ª « ¬ ­ © ® ¯ ° ± ² 2. Show its images on another complex plane, which we call ³ plane. Use ³ © ´ µ ¨ ¶ © · « ¬ ¸ © ¹ º » ¼ . Example 2.3 ½ ¾ ¿ À .
  • 12.
    8 Chapter 2Functions of Complex Variables Mapping of Á  10.80.60.40.20-0.2-0.4-0.6 2 1.5 1 0.5 0 Mapping Á  1. A straight line Ã Ä Å maps to a parabola Æ Ç È É Ê Ë Ì Í Î Ï Ë Ì Ð 2. A straight line Ñ Ò Ë maps to a parabola Ó Ì Ò Ê Ë Ì Í Î Ï Ë Ì Ð 3. A half circle given by Ô Ò Õ Ö × Ø Ù where Ú Û Ü Û Ý maps to a full circle given by Þ Ò Õ Ì Ö × Ø Ì Ù ß This also means that the upper half plane maps on to the entire complex plane. 4. A hyperbola à Ì Ï Ñ Ì Ò á maps to a straight line â ã á ä Mappings by Elementary Functions. 1. Translation by å æ is given by ç ã å è å æ ä 2. Rotation through an angle é æ is given by ç ã ê ë ì å ä 3. Rel• ection through x axis is given by ç ã í î ï 4. Exponential Function
  • 13.
    Mappings by ElementaryFunctions. 9 Exponential Function A vertical line maps to a circle. A horizontal line maps to a radial line. A horizontal strip enclosed between ð ñ ò and ð ñ ó ô maps to the entire complex plane. 5. Sine Function õ ö ÷ ø ñ õ ö ÷ ù ú û õ ü ð ý þ ú û õ ù õ ö ÷ ü ð A vertical line maps to a branch of a hyperbola. A horizontal line maps to an ellipse and has a period of ó ô .
  • 14.
    3 Analytic Functions Limits Afunction ÿ is de• ned in a deleted nbd of   ¡ ¢ De• nition 3.1 The limit of the function ÿ £   ¤ as   ¥ ¦ § is a number ¨ § if, for any given © there exists a such that ! # $ % ' ( ! ) 0 1 Example 3.1 # $ % 2 3 4 5 Show that 6 7 8 9 @ A B C D E F G H E I P Example 3.2 C D E F G E Q P Show that R S T U V U B C D E F G E Q I P Example 3.3 C D E F G E W XE P Show that the limit of C does not exist as E Y ` . Theorem 3.1 Let C D E F G a D b c d F e f g D b c d F and h I G a I e f g I P R S T U V U B C D E F G h I if and only if R S T i p q r s V i p B q r B s a G a I and R S T i p q r s V i p B q r B s g G g I P Example 3.4 C D E F G t S u E . Show that the R S T U V U B C D E F G t S u E I Example 3.5 C D E F G v b e f d Q . Show that the R S T U V Q w C D E F G x f . Theorem 3.2 If R S T U V U B C D E F G h I and R S T U V U B y D E F G €  ‚ ƒ „ … † ‡ † ˆ ‰  ‘ ’ “ ” • ‘ ’ “ – — ˜ ™ d e ™ f g h i j k j l m n o p q n o p — ˜ ™ e ™ f g h i j k j l m n o p r q n o p — ˜ ™ e ™ e ™ s— t u This theorem immediately makes available the entire machinery and tools used for real analysis to be applied to complex analysis. The rules for • nding limits then can be listed as follows:
  • 15.
    12 Chapter 3Analytic Functions 1. v w x y z y { | } | ~ 2. v w x y z y {  € }  €  ~ 3. v w x y z y { ‚ ƒ  „ } ‚ ƒ   „ if ‚ is a polynomial in  . 4. v w x y z y { … † ‡ ƒ  „ } … † ‡ ƒ   „ ~ 5. v w x y z y { ˆ w ‰ ƒ  „ } ˆ w ‰   ~ Continuity De• nition 3.2 A function Š , de•ned in some nbd of   is continuous at   if v w x y z y { Š ƒ  „ } Š ƒ   „ ~ This de• nition clearly assumes that the function is de• ned at   and the limit on the LHS exists. The function Š is continuous in a region if it is continuous at all points in that region. If funtions Š and ‹ are continuous at Œ  then Ž  ‹ , Ž ‹ and Ž  ‹ ‘ ‹ ‘ Œ  ’ “ ” • ’ are also continuous at Œ  . If a function Ž ‘ Œ ’ ” – ‘ — ˜ ™ ’  š › ‘ — ˜ ™ ’ is continuous at Œ  then the component functions – and › are also continuous at ‘ —  ˜ ™  ’ . Derivative De• nition 3.3 A function Ž , de•ned in some nbd of Œ  is differentiable at Œ  if œ  ž Ÿ   Ÿ ¡ Ž ‘ Œ ’ ¢ Ž ‘ Œ  ’ Œ ¢ Œ  exists. The limit is called the derivative of Ž at Œ  and is denoted by Ž £ ¤ ¥ ¦ § or ¨ © ª « ¬ ­ ® ¯ . Example 3.6 ° ¬ ­ ¯ ± ­ ² ³ Show that ° ´ ¬ ­ ¯ ± µ ­ ³ Example 3.7 ° ¬ ­ ¯ ± ¶ ­ ¶ ² . Show that this function is differentiable only at ­ ± · ¸ . In real analysis ¹ º ¹ is not differentiable but ¹ º ¹ » is. If a function is differentiable at ¼ , then it is continuous at ¼ .
  • 16.
    Cauchy-Riemann Equations 13 Theconverse in not true. See Example 3.7. Even if component functions of a complex function have all the partial derivatives, does not imply that the complex function will be differentiable. See Example 3.7. Some rules for obtaining the derivatives of functions are listed here. Let ½ and ¾ be differentiable at ¿ À 1. Á Á Â Ã ½ Ä ¾ Å Ã ¿ Å Æ ½ Ç Ã ¿ Å Ä ¾ Ã ¿ Å À 2. Á Á Â Ã ½ ¾ Å Ã ¿ Å Æ ½ Ç Ã ¿ Å ¾ Ã ¿ Å È ½ Ã ¿ Å ¾ Ç Ã ¿ Å À 3. Á Á Â Ã ½ É ¾ Å Ã ¿ Å Æ Ã ½ Ç Ã ¿ Å ¾ Ã ¿ Å Ê ½ Ã ¿ Å ¾ Ç Ã ¿ Å Å É Ë ¾ Ã ¿ Å Ì Í if ¾ Ã ¿ Å ÎÆ Ï À 4. Á Á Â Ã ½ Ð Ñ Ò Ó Ô Ò Õ Ö × Ó Ñ Ó Ô Ò Ò Ñ × Ó Ô Ò Ø 5. Ù Ù Ú Û Õ Ü Ø 6. Ù Ù Ú Ô Ý Õ Þ Ô Ý ß à Ø Cauchy-Riemann Equations Theorem 3.3 If Ö × Ó Ô á Ò exists, then all the •rst order partial derivatives of component function â Ó ã ä å Ò and æ Ó ã ä å Ò exist and satisfy Cauchy-Riemann Conditions: â ç Õ æ è â è Õ é æ ç Ø Example 3.8 Ö Ó Ô Ò Õ Ô ê Õ ã ê é å ê ë ì í ã å Ø Show that Cauchy-Riemann Condtions are satis•ed. Example 3.9 Ö Ó Ô Ò Õ î Ô î ê Õ ã ê ë å ê Ø Show that the Cauchy-Riemann Condtions are satis•ed only at Ô Õ Ü . Theorem 3.4 Let Ö Ó Ô Ò Õ â Ó ã ä å Ò ë ì æ Ó ã ä å Ò be de•ned in some nbd of the point Ô á Ø If the •rst partial derivatives of â and æ exist and are continuous at Ô á and satisfy Cauchy- Riemann equations at Ô á , then Ö is differentiable at Ô á and Ö × Ó Ô Ò Õ â ç ë ì æ ç Õ æ è é ì â è Ø Example 3.10 Ö Ó Ô Ò Õ ï ð ñ Ó Ô Ò Ø Show that Ö × Ó Ô Ò Õ ï ð ñ Ó Ô Ò Ø Example 3.11 Ö Ó Ô Ò Õ ò ó ô Ó Ô Ò Ø Show that Ö × Ó Ô Ò Õ õ ö ò Ó Ô Ò Ø Example 3.12 Ö Ó Ô Ò Õ ÷ ø ù ø ú Show that the CR conditions are satis•ed at û ü ý but the function þ is not differentiable at ý ú
  • 17.
    14 Chapter 3Analytic Functions If we write ÿ   ¡ ¢ £ ¤ then we can write Cauchy-Riemann Conditions in polar coordi- nates: ¥ ¦   § ¡ ¨ ¤ ¥ ¤   © ¡ ¨ ¦ Analytic Functions De• nition 3.4 A function is analytic in an open set if it has a derivative at each point in that set. De• nition 3.5 A function is analytic at a point ÿ if it is analytic in some nbd of ÿ . De• nition 3.6 A function is an entire function if it is analytic at all points of Example 3.13 ÿ   § ÿ is analytic at all nonzero points. Example 3.14 ÿ   ÿ is not analytic anywhere. A function is not analytic at a point ÿ but is analytic at some point in each nbd of ÿ then ÿ is called the singular point of the function . Harmonic Functions De• nition 3.7 A real valued function ! # $ % is said to be ' ( ) 0 1 2 3 in a domain of xy plane if it has continuous partial derivatives of the •rst and second order and satis•es 4 5 6 7 8 9 @ @ A B C D E F G : H I I P Q R S T U H V P Q R S T W X Y Theorem 3.5 If a function ` P a T W b P Q R S T U c d P Q R S T is analytic in a domain e then the functions f and g are harmonic in e . De• nition 3.8 If two given functions f h i p q r and g h i p q r are harmonic in domain D and their •rst order partial derivatives satisfy Cauchy-Riemann Conditions f s t g u f u t v g s w then g is said to be x y €  ‚ ƒ „ … … ‚ ƒ † ‡ ˆ ‰  ‘ of ’ “ Example 3.15 Let ’ ” • – — ˜ ™ • d e — d and f ” • – — ˜ ™ g • — “ Show that f is hc of ’ and not vice versa. Example 3.16 ’ ” • – — ˜ ™ — h e i • — . Find harmonic conjugate of ’ “
  • 18.
    4 Integrals Contours Example 4.1Represent a line segment joining points j k l m n and o p q r r n by parametric equations. Example 4.2 Show that a half circle in upper half plane with radius s and centered at origin can be parametrized in various ways as given below: 1. t u v w x s y z { v | } u v w x s { ~  v | where v €  ‚ ƒ „ 2. … † ‡ ˆ ‰ ‡ Š ‹ † ‡ ˆ ‰ Œ  Ž  ‡ Ž Š where ‡ €   ‚  „ 3. … † ‡ ˆ ‰  †  ‡  ‘ ˆ Š ‹ † ‡ ˆ ‰   Œ ‡  ‡ Ž Š where ‡ €  ‚ ‘ „ De• nition 4.1 A set of points ’ ‰ † … Š ‹ ˆ in complex plane is called an “ ” • if … ‰ … † ‡ ˆ Š ‹ ‰ ‹ † ‡ ˆ Š † – — ‡ — ˜ ˆ where … † ‡ ˆ and ‹ † ‡ ˆ are continuous functions of the real parameter ‡ „ Example 4.3 … † ‡ ˆ ‰ ™ š › ‡ Š ‹ † ‡ ˆ ‰ › œ  †  ‡ ˆ , where ‡ €  ‚  žƒ . Show that the curve cuts itself and is closed. An arc is called simple if ‡ Ÿ  ‰ ‡ Ž ¡ ’ † ‡ Ÿ ˆ  ‰ ’ † ‡ Ž ˆ „ An arc is closed if ’ † – ˆ ‰ ’ † ˜ ˆ . An arc is differentiable if ’ ¢ † ‡ ˆ ‰ … ¢ † ‡ ˆ £ ¤ ‹ ¢ † ‡ ˆ exists and … ¢ † ‡ ˆ and ‹ ¢ † ‡ ˆ are continu- ous. A smooth arc is differentiable and ’ ¢ † ‡ ˆ is nonzero for all ‡ . De• nition 4.2 Length of a smooth arc is de•ned as ¥ ¦ § ¨ © ª « ¬ ­ ® ¯ ° ± ² ³ ´ µ ² ¶ The length is invariant under parametrization change.
  • 19.
    16 Chapter 4Integrals De• nition 4.3 A · ¸ ¹ º ¸ » ¼ is a constructed by joining •nite smooth curves end to end such that ½ ¾ ¿ À is continuous and ½ Á ¾ ¿ À is piecewise continuous. A closed simple contour has only • rst and last point same and does not cross itself. Contour Integral If  is a contour in complex plane de• ned by ½ ¾ ¿ À Ã Ä ¾ ¿ À Å Æ Ç ¾ ¿ À and a function È ¾ ½ À Ã É ¾ Ä Ê Ç À Å Æ Ë ¾ Ä Ê Ç À is de• ned on it. The integral of È ¾ ½ À along the contour  is denoted and de• ned as follows: Ì Í Î Ï Ð Ñ Ò Ð Ó Ô Õ Ö Î Ï Ð Ñ Ð × Ï Ø Ñ Ò Ø Ó Ô Õ Ö Ï Ù Ú × Û Ü Ý × Ñ Ò Ø Þ ß Ô Õ Ö Ï Ù Ý × Þ Ü Ú × Ñ Ò Ø Ó Ô Ï Ù Ò Ú Û Ü Ò Ý Ñ Þ ß Ô Ï Ù Ò Ý Þ Ü Ò Ú Ñ The component integrals are usual real integrals and are well de• ned. In the last form appropriate limits must placed in the integrals. Some very straightforward rules of integration are given below: 1. à á â ã ä å æ ç å è â à á ã ä å æ ç å where â is a complex constant. 2. à á ä ã ä å æ é ê ä å æ æ ç å è à á ã ä å æ ç å é à á ê ä å æ ç å ë 3. à á ì í á î ã ä å æ ç å è à á ì ã ä å æ ç å é à á î ã ä å æ ç å ë 4. ï ï ð ñ ò ó ô õ ö ô ï ï ÷ ð ñ ø ò ó ô ó ù õ õ ô ú ó ù õ ø ö ù û 5. If ø ò ó ô õ ø ÷ ü for all ý þ ÿ then     ¡ ¢ £ ¤ ¥ ¦ § ¥     ¨ © , where is length of the countour ÿ Example 4.4 £ ¤ ¥ ¦ ¥ Find integral of £ from ¤ ¦ to ¤ ¦ along a straight line and also along st line path from ¤ ¦ to ¤ ¦ and from ¤ ¦ to ¤ ¦ . Example 4.5 £ ¤ ¥ ¦ ¥ . Find the integral from ¤ ¦ to ¤ ¦ along a semicircu- lar path in upper plane given by ¥ Example 4.6 Show that     ¡ ¢ £ ¤ ¥ ¦ § ¥     ¨ ! for £ ¤ ¥ ¦ # ¥ $ and ÿ % ¥ from to
  • 20.
    Cauchy-Goursat Theorem 17 Cauchy-GoursatTheorem Theorem 4.1 (Jordan Curve Theorem) Every simple and closed contour in complex plane splits the entire plane into two domains one of which is bounded. The bounded domain is called the interior of the countour and the other one is called the exterior of the contour. De• ne a sense direction for a contour. Theorem 4.2 Let ' be a simple closed contour with positive orientation and let ( be the interior of ' ) If 0 and 1 are continuous and have continuous partial derivatives 2 3 4 2 5 4 1 3 and 1 5 at all points on 6 and 7 , then 8 9 @ 2 @ A 4 B C D A E 1 @ A 4 B C D B C F 8 8 G 1 3 @ A 4 B C H 2 5 @ A 4 B C I D A D B Theorem 4.3 (Cauchy-Goursat Theorem) Let P be analytic in a simply connected domain 7 Q If 6 is any simple closed contour in 7 , then 8 9 P @ R C D R F S Q Example 4.7 P @ R C F R T 4 U V W @ R C 4 X Y ` @ R C etc are entire functions so integral about any loop is zero. Theorem 4.4 Let 6 a and 6 T be two simple closed positively oriented contours such that 6 T lies entirely in the interior of 6 a Q If P is an analytic function in a domain 7 that contains 6 a and 6 T both and the region between them, then 8 9 b P @ R C D R F 8 9 c P @ R C D R Q Example 4.8 P @ R C F d e R . Find f 9 P @ R C D R if 6 is any contour containing origin. Choose a circular contour inside 6 Q Example 4.9 f 9 a g h g i D R F p q r if 6 contains R s Q Example 4.10 Find f 9 T g t g g c u T where 6 v w R w F p Q Extend the Cauchy Goursat theorem to multiply connected domains. Antiderivative Theorem 4.5 (Fundamental Theorem of Integration) Let P be de•ned in a simply connected domain 7 and is analytic in 7 . If R s and R are points in 7 and 6 is any contour in 7 joining R s and R 4 then the function x @ R C F 8 9 P @ R C D R
  • 21.
    18 Chapter 4Integrals is analytic in y and €  ‚ ƒ „ … † ‚ ƒ „ ‡ De• nition 4.4 If † is analytic in y and ƒ ˆ and ƒ ‰ are two points in y then the de•nite integral is de•ned as  ‘ ’  “ † ‚ ƒ „ ” ƒ … € ‚ ƒ ‰ „ • € ‚ ƒ ˆ „ where € is an antiderivative of † . Example 4.11 ‰ – — ˜ ™ ƒ ‰ ” ƒ … ƒ d e f g g ‰ – — ™ … ‰ d h i ˆ ˆ d ‡ Example 4.12 — ˜ ˆ j k l ƒ … l m n ƒ o — ˆ … l m n i • l m n p ‡ Example 4.13  ‘ ˜  “ q   … r k s ƒ ‰ • r k s ƒ ˆ ‡ Cauchy Integral Formula Theorem 4.6 (Cauchy Integral Formula) Let † be analytic in domain y . Let t be a positively oriented simple closed contour in y . If ƒ ™ is in the interior of t then † ‚ ƒ ™ „ … p u v i ’ w † ‚ ƒ „ ” ƒ ƒ • ƒ ™ ‡ Example 4.14 † ‚ ƒ „ … ˆ  ‘ – x ‡ Find ˜ w † ‚ ƒ „ ” ƒ if t y o ƒ • i o … u ‡ Example 4.15 † ‚ ƒ „ …  ‰  – ˆ ‡ Find ˜ w † ‚ ƒ „ ” ƒ if t is square with vertices on ‚ z u { z u „ ‡ Theorem 4.7 If † is analytic at a point, then all its derivatives exist and are analytic at that point. † | } ~ ‚ ƒ ™ „ …  € u v i ’ w † ‚ ƒ „ ” ƒ ‚ ƒ • ƒ ™ „ } – ˆ
  • 22.
    5 Series Convergence ofSequences and Series Example 5.1  ‚ ƒ „ … † Example 5.2  ‚ ƒ ‡ ‚ Example 5.3  ‚ ƒ ˆ ‰ Š ‹ Œ ˆ „  ‚ … † Ž De• nition 5.1 An in•nite sequece     Ž  ‘ ‘ ‘   ‚  ‘ ‘ ‘ of complex numbers has a ’ “ ” “ •  if for each positive – , there exists positive integer — such that ˜ ™ š › ™ ˜ œ  ž Ÿ   ¡   ¢ £ ¤ ¥ ¦ § ¨ The sequences have only one limit. A sequence said to converge to © if © is its limit. A sequence diverges if it does not converge. Example 5.4 © ª « © ª converges to 0 if ¬ © ¬ ­ ® else diverges. Example 5.5 © ª « ® ¯ ° ¥ ± ² ³ ¥ ± ® ´ ¯ ¥ converges to ² . Theorem 5.1 Suppose that © ª « µ ª ± ² ¶ ª and © « µ ± ² ¶ ¨ Then, · ¸ ¹ ª º » ¼ ½ ¾ ¼ if and only if ¿ À Á ½  » à ½ ¾ Ã Ä Å Æ Ç È É Ê Ë Ì Í Ê Î Í De• nition 5.2 If Ï Ð Ê Ñ is a sequence, the in•nite sum Ð Ò Ó Ð Ô Ó Õ Õ Õ Ó Ð Ê Ó Õ Õ Õ is called a series and is denoted by Ì Ö × Ø Ù Ú Û . De• nition 5.3 A series Ü Ý Û Ø Ù Ú Û is said to converge to a sum Þ if a sequence of partial sums Þ ß à á â ã á ä ã å å å ã á æ converges to ç .
  • 23.
    20 Chapter 5Series Theorem 5.2 Suppose that è é ê ë é ì í î é and ï ê ð ñ ò ó ô Then, õ ö ÷ ø ù ú ÷ û ü if and only if ý ö ÷ ø ù þ ÷ û ÿ   ¡ ¢ ý ö ÷ ø ù £ ÷ û ¤ Example 5.6 ¥ ý ¦ ú ÷ û § ¨ © § ú if ú § Taylor Series Theorem 5.3 (Taylor Series) If is analytic in a circular disc of radius ¦ andcentered at ú ¦ then at each point inside the disc there is a series representation for given by © ú û ý ö ÷ ø ¦ ÷ © ú ú ¦ ÷ where ÷ û ÷ © ú ¦ ! Example 5.7 # ¡ ú û ¥ ý ¦ ú $ ÷ % ù ¨ © ' § ! ( Example 5.8 ù ù % ) û ¥ ý ¦ © § ÷ ú ÷ ú § ( Example 5.9 0 ) û ¥ ý ¦ ú ÷ ¨ ! ( Example 5.10 ù ) û ¥ ý ¦ © § ÷ © ú § ÷ ú § § ( Laurent Series Theorem 5.4 (Laurent Series) If is analytic at all points ú in an annular region 1 such that ù ú ú ¦ $ then at each point in 1 there is a series representation for given by © ú û ý ö ÷ ø ¦ ÷ © ú ú ¦ ÷ ' ý ö ÷ ø ù 2 ÷ © ú ú ¦ ÷ where ÷ û § 3 4 5 6 © ú 7 ú © ú ú ¦ ÷ % ù 2 ÷ û § 3 4 5 6 © ú 7 ú © ú ú ¦ 8 ÷ % ù
  • 24.
    Laurent Series 21 and9 is any contour in @ . Example 5.11 If A is analytic inside a disc of radius B about C D E then the Laurent series for A is identical to the Taylor series for A . That is all F G H I . Example 5.12 P P Q R H S T G U P V W P X Y ` a b c where d e d f g h Example 5.13 i p e q r s t u b s t v u b s w v h Find Laurent series for all d e d x g y g x d e d x € and d e d f € h Example 5.14 Note that  t r t w ‚ ƒ „ … i p e q † e h
  • 25.
    6 Theory ofResidues And Its Applications Singularities De• nition 6.1 If a function ‡ fails to be analytic at ˆ ‰ but is analytic at some point in each neighbourhood of ˆ ‰ , then ˆ ‰ is a  ‘ ’ “ ” • – — ˜ ™ ‘ ’ d of e . De• nition 6.2 If a function e fails to be analyitc at f g but is analytic at each f in h i j f k f g j i l for some l m then e is said to be an ‘  ™ • – d n o p q r s t u v w x y q r z of { . Example 6.1 { | } ~  €  } has an isolated singularity at ‚ . Example 6.2 { | } ~  €  ƒ „ … | † } ~ has isolated singularities at }  ‚ ‡ ˆ € ‡ ‰ ‰ ‰ . Example 6.3 { | } ~  €  ƒ „ … | †  } ~ has isolated singularities at }  €  Š for integral Š , also has a singularity at }  ‚ . Example 6.4 { | } ~  ‹ Œ  } all points of negative x-axis are singular. Types of singularities If a function { has an isolated singularity at } Ž then  a  such that ‘ is analytic at all points in ’ “ ” • – • — ” “  . Then ‘ must have a Laurent series expansion about • — . The part ˜ ™š › œ  œ ž • – • — Ÿ   š is called the principal part of ‘ ¡ 1. If there are in• nite nonzero  ¢ in the prinipal part then • — is called an essential singu- larity of ‘ ¡ 2. If for some integer £ ¤ ¥ ¦ § ¨ © but ª « ¨ © for all ¬ ­ ® then ¯ ° is called a pole of order ® of ± ² If ® ¨ ³ then it is called a simple pole. 3. If all ª s are zero then ¯ ° is called a removable singularity.
  • 26.
    24 Chapter 6Theory of Residues And Its Applications Example 6.5 ´ µ ¶ · ¸ µ ¹ º » ¶ · ¼ ¶ is unde•ned at ¶ ¸ ½ ¾ ´ has a removable isolated singularity at ¶ ¸ ½ ¾ Residues Suppose a function ´ has an isolated singularity at ¶ ¿ À then there exists a Á Â ½ such that ´ is analytic for all ¶ in deleted nbd ½ Ã Ä ¶ Å ¶ ¿ Ä Ã Á . Then ´ has a Laurent series representation ´ µ ¶ · ¸ Æ Ç È É ¿ Ê È µ ¶ Å ¶ ¿ · È Ë Æ Ç È É Ì Í È µ ¶ Å ¶ ¿ · È ¾ The coef• cient Í Ì ¸ Î Ï Ð Ñ Ò Ó ´ µ ¶ · Ô ¶ where Õ is any contour in the deleted nbd, is called the residue of ´ at ¶ ¿ ¾ Example 6.6 ´ µ ¶ · ¸ Ì Ö × Ö Ø . Then Ù Ó ´ µ ¶ · Ô ¶ ¸ Ï Ð Ñ Í Ì ¸ Ï Ð Ñ if Õ contains ¶ ¿ , other- wise ½ . Example 6.7 ´ µ ¶ · ¸ Ì Ö Ú Ö × Ö Ø Û Ü Ý Show Þ ß à á â ã ä â å æ ç è é if ê ë ì í î ï ì ð ñ ò Example 6.8 ó ô í õ ð í ö ÷ ø ù ú û ü . Show ý þ ó ô í õ ÿ í ð   ¡ if ê ë ì í ì ð ñ ò Example 6.9 ó ô í õ ð ö ÷ ø ù ú û ¢ ü . Show ý þ ó ô í õ ÿ í ð £ if ê ë ì í ì ð ñ even though it has a singularity at í ð £ ò Theorem 6.1 If a function ó is analytic on and inside a positively oriented countour ê , except for a •nite number of points í ú ¤ í ¥ ¦ ò ò ò ¦ í § inside ¨ , then © § ! # Res ! $ Example 6.10 Show that % ' ( ) ' 0 ' ( 1 2 3 4 5 6 7 8 9 Residues of Poles Theorem 6.2 If a function @ has a pole of order A at 4 B then Res@ C 4 B D 5 E F G ' H ' I P Q R S T Q U V W X Y ` W a X Y ` R R a T a b U X c R a U U d e Example 6.11 f g h i p q r s r t e Simple pole Res f g h u i p v e
  • 27.
    Quotients of AnalyticFunctions 25 Example 6.12 w x y €  ‚ ƒ „ ƒ … ƒ † ‡ ˆ . Simple pole at y  ‰  Res w x ‰ €  ‘ ’ ‘ “ . Pole of order ” at y  • . Res w x • €  – ‘ ’ ‘ “ . Example 6.13 w x y €  — ˜ ™ d d e f d g h i j . Res k l m n o p q r s t . Res k l s n o p u v p s w x r y s t z Quotients of Analytic Functions Theorem 6.3 If a function k l { n o | } ~  €  ‚ ƒ , where „ and … are analytic at † ‡ ˆ then 1. ‰ is singular at † ‡ iff … Š † ‡ ‹ Œ  Ž 2. ‰ has a simple pole at † ‡ if …  Š † ‡ ‹ Œ  . Then residue of ‰ at † ‡ is „ Š † ‡ ‹ ‘ …  Š † ‡ ‹ .
  • 28.
    A References This appendixcontains the references.
  • 29.
    B Index This appendixcontains the index.