This document provides information about the theoretical physics course Phys2325 at the University of Hong Kong, including the course code, homepage, lecturer contact information, textbook, contents covered, and assessment details. The course covers complex variables and their applications in theoretical physics, including Cauchy's integral formula, properties of special functions, Fourier series, and solutions to partial differential equations. Students will be assessed through one 3-hour written exam worth 80% and a course assessment worth 20% of the final grade.
Mathematics and History of Complex VariablesSolo 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
The document discusses analytic functions and the Cauchy-Riemann equations. It provides examples of determining whether functions are analytic or not based on satisfying the Cauchy-Riemann equations. It also discusses harmonic functions and their conjugate harmonic functions. Specifically, it defines analytic and harmonic functions, presents the Cauchy-Riemann equations and examples of applying them, discusses harmonic functions and provides examples of determining if functions are harmonic.
The document discusses complex differentiability and analytic functions. It shows that for a function f(z) to be complex differentiable, the Cauchy-Riemann equations relating the partial derivatives of the real and imaginary parts must be satisfied. It also discusses representing functions as power series and their radii of convergence. Multivalued functions like logarithms and roots are discussed, noting the need for branch cuts to define single-valued branches.
This document discusses complex variables and functions. It covers topics such as:
- Cauchy-Riemann conditions, which must be satisfied for a complex function to be analytic/differentiable
- Cauchy's integral theorem, which states that the integral of an analytic function around a closed contour is zero
- Harmonic functions, whose real and imaginary parts satisfy the 2D Laplace equation
- Examples of calculating contour integrals of functions like zn along circular and square paths
The document provides proofs of theorems like Cauchy's integral theorem using techniques like Stokes' theorem. It also discusses simply and multiply connected regions in relation to contour integrals.
This document provides examples of transformations involving complex variables and their applications. It contains 3 examples of inversion transformations where a line or circle in the z-plane is transformed to a circle or line in the w-plane. It also contains 2 examples of square transformations where a region in the z-plane is transformed to parabolic regions in the w-plane. Additionally, it discusses finding the image of a line or circle under translations in the complex plane.
This document provides 22 questions related to complex integration and contour integration. It covers key concepts like Cauchy's integral theorem, Cauchy's integral theorem for derivatives, evaluating contour integrals, Taylor series expansions, Laurent series expansions, isolated singular points, essential singular points, removable singular points, finding poles and residues, and applying Cauchy's residue theorem. The questions are divided into two parts - the first part focuses on definitions and properties, while the second part involves applying these concepts to evaluate specific contour integrals.
- The document discusses various types of power series representations of complex functions, including Taylor series, Maclaurin series, and Laurent series.
- It defines key concepts such as isolated singularities, classification of singularities into removable, pole, and essential types based on the principal part of the Laurent series, and the residue, which is the coefficient of the 1/(z-z0) term in the Laurent series at an isolated singularity z0.
- Examples are provided to illustrate these various types of series representations and singularities.
Mathematics and History of Complex VariablesSolo 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
The document discusses analytic functions and the Cauchy-Riemann equations. It provides examples of determining whether functions are analytic or not based on satisfying the Cauchy-Riemann equations. It also discusses harmonic functions and their conjugate harmonic functions. Specifically, it defines analytic and harmonic functions, presents the Cauchy-Riemann equations and examples of applying them, discusses harmonic functions and provides examples of determining if functions are harmonic.
The document discusses complex differentiability and analytic functions. It shows that for a function f(z) to be complex differentiable, the Cauchy-Riemann equations relating the partial derivatives of the real and imaginary parts must be satisfied. It also discusses representing functions as power series and their radii of convergence. Multivalued functions like logarithms and roots are discussed, noting the need for branch cuts to define single-valued branches.
This document discusses complex variables and functions. It covers topics such as:
- Cauchy-Riemann conditions, which must be satisfied for a complex function to be analytic/differentiable
- Cauchy's integral theorem, which states that the integral of an analytic function around a closed contour is zero
- Harmonic functions, whose real and imaginary parts satisfy the 2D Laplace equation
- Examples of calculating contour integrals of functions like zn along circular and square paths
The document provides proofs of theorems like Cauchy's integral theorem using techniques like Stokes' theorem. It also discusses simply and multiply connected regions in relation to contour integrals.
This document provides examples of transformations involving complex variables and their applications. It contains 3 examples of inversion transformations where a line or circle in the z-plane is transformed to a circle or line in the w-plane. It also contains 2 examples of square transformations where a region in the z-plane is transformed to parabolic regions in the w-plane. Additionally, it discusses finding the image of a line or circle under translations in the complex plane.
This document provides 22 questions related to complex integration and contour integration. It covers key concepts like Cauchy's integral theorem, Cauchy's integral theorem for derivatives, evaluating contour integrals, Taylor series expansions, Laurent series expansions, isolated singular points, essential singular points, removable singular points, finding poles and residues, and applying Cauchy's residue theorem. The questions are divided into two parts - the first part focuses on definitions and properties, while the second part involves applying these concepts to evaluate specific contour integrals.
- The document discusses various types of power series representations of complex functions, including Taylor series, Maclaurin series, and Laurent series.
- It defines key concepts such as isolated singularities, classification of singularities into removable, pole, and essential types based on the principal part of the Laurent series, and the residue, which is the coefficient of the 1/(z-z0) term in the Laurent series at an isolated singularity z0.
- Examples are provided to illustrate these various types of series representations and singularities.
This document discusses functions of a complex variable. It introduces complex numbers and their representations. It covers topics like complex differentiation using Cauchy-Riemann equations, analytic functions, Cauchy's integral theorem, and contour integrals. Functions of a complex variable provide tools for physics concepts involving complex quantities like wavefunctions. Cauchy's integral theorem states that the contour integral of an analytic function over a closed path is zero.
The document discusses various topics related to complex functions and complex analysis. It defines concepts such as distance between complex numbers, circles, circular disks, neighborhoods, annuli, open and closed sets, connected sets, domains, regions, bounded regions, single-valued and multi-valued functions, and limits and continuity of complex functions. Specific examples are provided to illustrate definitions of circles, neighborhoods, single-valued and multi-valued functions. The limit of a complex function as z approaches a point z0 is defined using the epsilon-delta definition of a limit.
Cauchy integral theorem & formula (complex variable & numerical method )Digvijaysinh Gohil
1) The document discusses the Cauchy Integral Theorem and Formula. It states that if a function f(z) is analytic inside and on a closed curve C, then the integral of f(z) around C is equal to 0.
2) It provides examples of evaluating integrals using the Cauchy Integral Theorem when the singularities lie outside the closed curve C.
3) The Cauchy Integral Formula is introduced, which expresses the value of an analytic function F(a) inside C as a contour integral around C. Examples are worked out applying this formula to find the value and derivatives of functions at points inside C.
1. The document introduces complex numbers and some basic results regarding complex numbers such as the complex conjugate and modulus of a complex number.
2. It then discusses functions of a complex variable, defining a complex function and its Cartesian and polar forms. It also covers continuity, derivatives, and analytic functions of a complex variable.
3. The Cauchy-Riemann equations are derived and provide a necessary condition for a function to be analytic (differentiable everywhere in a neighborhood). Two examples are provided to illustrate the Cauchy-Riemann equations and analytic functions.
1. The document provides 14 problems involving partial differential equations (PDEs). The problems involve forming PDEs by eliminating arbitrary constants from functions, finding complete integrals, and solving PDEs.
2. Methods used include taking partial derivatives, finding auxiliary equations, and making substitutions to isolate the PDE or solve it.
3. The document covers a range of techniques for working with PDEs, including eliminating constants, finding trial solutions, integrating subsidiary equations, and solving auxiliary equations to find complete integrals.
- The document discusses complex analytic functions and their derivatives.
- A complex function is analytic if it has a complex derivative everywhere in some open region of the complex plane. This allows drawing richer conclusions than for real differentiable functions.
- The derivative of a complex function f at z0 is defined as the limit of (f(z)-f(z0))/(z-z0) as z approaches z0. If this limit exists, f is said to be differentiable or analytic at z0.
Complex analysis and differential equationSpringer
This document introduces holomorphic functions and some of their key properties. It begins by defining limits, continuity, differentiability, and holomorphic functions. It then introduces the Cauchy-Riemann equations, which provide a necessary condition for differentiability involving the partial derivatives of the real and imaginary parts. Several examples are provided to illustrate these concepts. The document also discusses properties of derivatives of holomorphic functions and proves that differentiability implies continuity. It concludes by defining connected sets.
1. The document discusses various complex transformations and their properties. It defines conformal transformations as those that map curves to curves while preserving angles.
2. Several examples of transformations are analyzed, including w = 1/z, w = z^2, and the Joukowski transformation w = z + 1/z. It is shown that circles map to circles or lines under w = 1/z, and that conformal mappings preserve orthogonality of curves.
3. Bilinear transformations of the form w = (az + b)/(cz + d) are introduced, where ad - bc ≠ 0. Such transformations map points to points and have two fixed or invariant points.
This unit covers the formation and solutions of partial differential equations (PDEs). PDEs can be obtained by eliminating arbitrary constants or functions from relating equations. Standard methods are used to solve first order PDEs and higher order linear PDEs with constant coefficients. Various physical processes are modeled using PDEs including the wave equation, heat equation, and Laplace's equation.
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
The document discusses line integrals in the complex plane. It defines line integrals, shows how complex line integrals are equivalent to two real line integrals, and reviews how to parameterize curves to evaluate line integrals. It also covers Cauchy's theorem, which states that the line integral of an analytic function around a closed curve in its domain is zero. The fundamental theorem of calculus for complex variables and the Cauchy integral formula are also summarized.
1. The document discusses various types of transformations in complex analysis, including translation, rotation, stretching, and inversion.
2. Under inversion (1/w=z), a straight line is mapped to a circle if it does not pass through the origin, and to another straight line if it does pass through the origin. A circle is always mapped to another circle.
3. A general bilinear or Möbius transformation can be expressed as a combination of translation, rotation, stretching, and inversion.
The document defines proper and improper integrals, and discusses different types of improper integrals based on whether the limits are infinite or the function is unbounded. It provides tests to determine if improper integrals converge or diverge, including the T1 test involving exponential functions, the T2 test involving power functions, and comparison tests. Examples are worked through applying these tests to determine if various improper integrals converge or diverge. The key information is on defining improper integrals and tests to analyze their convergence.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IRai University
1. The document discusses functions of complex variables, including analytic functions, Cauchy-Riemann equations, harmonic functions, and methods for determining an analytic function when its real or imaginary part is known.
2. Some key topics covered are the definition of an analytic function, Cauchy-Riemann equations in Cartesian and polar forms, properties of analytic functions including orthogonal systems, and determining the analytic function using methods like direct, Milne-Thomson's, and exact differential equations.
3. Examples are provided to illustrate determining the analytic function given its real or imaginary part, such as finding the function when the real part is a polynomial or the imaginary part is a trigonometric function.
1. The document contains an assignment on mathematics involving ordinary and partial differential equations, matrices, vector calculus, and their applications. It includes solving various types of differential equations, finding eigenvectors and eigenvalues of matrices, evaluating line, surface and volume integrals using Green's theorem and Stokes' theorem. The assignment contains 20 problems spanning these topics.
2. The assignment covers key concepts in differential equations, linear algebra, and vector calculus including solving ordinary differential equations, partial differential equations, systems of linear equations, eigenproblems, line integrals, surface integrals, divergence, curl, gradient, Laplacian, and theorems like Green's theorem and Stokes' theorem.
3. Students are required to solve 20 problems involving these
This document provides information about a theoretical physics course, including the course code, homepage, lecturer contact information, textbook, main topics covered, and assessment details. The course covers complex variables and their applications in theoretical physics. Topics include Cauchy's integral formula, calculus of residues, partial differential equations, special functions, and Fourier series. Students will be assessed through a 3-hour written exam worth 80% and a course assessment worth 20%.
The document discusses various types of differential equations including ordinary differential equations (ODEs) and partial differential equations (PDEs). It defines key terms like order, degree, and describes several methods for solving common types of differential equations, such as separating variables, exact differentials, linear equations, Bernoulli's equation, and Clairaut's equation. It also includes sample problems and solutions for each method and concludes with multiple choice questions.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IIRai University
This document provides an overview of Unit II - Complex Integration from the Engineering Mathematics-IV course at RAI University, Ahmedabad. It covers key topics such as:
1) Complex line integrals and Cauchy's integral theorem which states that the integral of an analytic function around a closed curve is zero.
2) Cauchy's integral formula which can be used to evaluate integrals and find derivatives of analytic functions.
3) Taylor and Laurent series expansions of functions, including their regions of convergence.
4) The residue theorem which can be used to evaluate real integrals involving trigonometric or rational functions.
The document provides an overview of functions of a complex variable. Some key points:
1) Functions of a complex variable provide powerful tools in theoretical physics for quantities that are complex variables, evaluating integrals, obtaining asymptotic solutions, and performing integral transforms.
2) The Cauchy-Riemann equations are a necessary condition for a function f(z) = u(x,y) + iv(x,y) to be differentiable at a point. If the equations are satisfied, the function is analytic.
3) Cauchy's integral theorem states that if a function f(z) is analytic in a simply connected region R, the contour integral of f(z) around any closed path in
This document discusses complex numbers and functions. It introduces complex numbers using Cartesian (x + iy) and polar (r(cosθ + i sinθ)) forms. It describes the Cauchy-Riemann conditions that must be satisfied for a function of a complex variable to be differentiable. A function is analytic if it satisfies the Cauchy-Riemann conditions and its partial derivatives are continuous. Analytic functions have properties like equality of second-order partial derivatives and establishing a relation between the real and imaginary parts.
1) Complex numbers can be represented in Cartesian (x + iy) or polar (r(cosθ + i sinθ)) form, with conversions between the two.
2) The derivative of a complex function f(z) is defined if the Cauchy-Riemann equations are satisfied.
3) A function is analytic if it is differentiable and its partial derivatives are continuous, implying the Cauchy-Riemann equations always hold. Analytic functions have properties like equality of second partial derivatives.
This document discusses functions of a complex variable. It introduces complex numbers and their representations. It covers topics like complex differentiation using Cauchy-Riemann equations, analytic functions, Cauchy's integral theorem, and contour integrals. Functions of a complex variable provide tools for physics concepts involving complex quantities like wavefunctions. Cauchy's integral theorem states that the contour integral of an analytic function over a closed path is zero.
The document discusses various topics related to complex functions and complex analysis. It defines concepts such as distance between complex numbers, circles, circular disks, neighborhoods, annuli, open and closed sets, connected sets, domains, regions, bounded regions, single-valued and multi-valued functions, and limits and continuity of complex functions. Specific examples are provided to illustrate definitions of circles, neighborhoods, single-valued and multi-valued functions. The limit of a complex function as z approaches a point z0 is defined using the epsilon-delta definition of a limit.
Cauchy integral theorem & formula (complex variable & numerical method )Digvijaysinh Gohil
1) The document discusses the Cauchy Integral Theorem and Formula. It states that if a function f(z) is analytic inside and on a closed curve C, then the integral of f(z) around C is equal to 0.
2) It provides examples of evaluating integrals using the Cauchy Integral Theorem when the singularities lie outside the closed curve C.
3) The Cauchy Integral Formula is introduced, which expresses the value of an analytic function F(a) inside C as a contour integral around C. Examples are worked out applying this formula to find the value and derivatives of functions at points inside C.
1. The document introduces complex numbers and some basic results regarding complex numbers such as the complex conjugate and modulus of a complex number.
2. It then discusses functions of a complex variable, defining a complex function and its Cartesian and polar forms. It also covers continuity, derivatives, and analytic functions of a complex variable.
3. The Cauchy-Riemann equations are derived and provide a necessary condition for a function to be analytic (differentiable everywhere in a neighborhood). Two examples are provided to illustrate the Cauchy-Riemann equations and analytic functions.
1. The document provides 14 problems involving partial differential equations (PDEs). The problems involve forming PDEs by eliminating arbitrary constants from functions, finding complete integrals, and solving PDEs.
2. Methods used include taking partial derivatives, finding auxiliary equations, and making substitutions to isolate the PDE or solve it.
3. The document covers a range of techniques for working with PDEs, including eliminating constants, finding trial solutions, integrating subsidiary equations, and solving auxiliary equations to find complete integrals.
- The document discusses complex analytic functions and their derivatives.
- A complex function is analytic if it has a complex derivative everywhere in some open region of the complex plane. This allows drawing richer conclusions than for real differentiable functions.
- The derivative of a complex function f at z0 is defined as the limit of (f(z)-f(z0))/(z-z0) as z approaches z0. If this limit exists, f is said to be differentiable or analytic at z0.
Complex analysis and differential equationSpringer
This document introduces holomorphic functions and some of their key properties. It begins by defining limits, continuity, differentiability, and holomorphic functions. It then introduces the Cauchy-Riemann equations, which provide a necessary condition for differentiability involving the partial derivatives of the real and imaginary parts. Several examples are provided to illustrate these concepts. The document also discusses properties of derivatives of holomorphic functions and proves that differentiability implies continuity. It concludes by defining connected sets.
1. The document discusses various complex transformations and their properties. It defines conformal transformations as those that map curves to curves while preserving angles.
2. Several examples of transformations are analyzed, including w = 1/z, w = z^2, and the Joukowski transformation w = z + 1/z. It is shown that circles map to circles or lines under w = 1/z, and that conformal mappings preserve orthogonality of curves.
3. Bilinear transformations of the form w = (az + b)/(cz + d) are introduced, where ad - bc ≠ 0. Such transformations map points to points and have two fixed or invariant points.
This unit covers the formation and solutions of partial differential equations (PDEs). PDEs can be obtained by eliminating arbitrary constants or functions from relating equations. Standard methods are used to solve first order PDEs and higher order linear PDEs with constant coefficients. Various physical processes are modeled using PDEs including the wave equation, heat equation, and Laplace's equation.
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
The document discusses line integrals in the complex plane. It defines line integrals, shows how complex line integrals are equivalent to two real line integrals, and reviews how to parameterize curves to evaluate line integrals. It also covers Cauchy's theorem, which states that the line integral of an analytic function around a closed curve in its domain is zero. The fundamental theorem of calculus for complex variables and the Cauchy integral formula are also summarized.
1. The document discusses various types of transformations in complex analysis, including translation, rotation, stretching, and inversion.
2. Under inversion (1/w=z), a straight line is mapped to a circle if it does not pass through the origin, and to another straight line if it does pass through the origin. A circle is always mapped to another circle.
3. A general bilinear or Möbius transformation can be expressed as a combination of translation, rotation, stretching, and inversion.
The document defines proper and improper integrals, and discusses different types of improper integrals based on whether the limits are infinite or the function is unbounded. It provides tests to determine if improper integrals converge or diverge, including the T1 test involving exponential functions, the T2 test involving power functions, and comparison tests. Examples are worked through applying these tests to determine if various improper integrals converge or diverge. The key information is on defining improper integrals and tests to analyze their convergence.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IRai University
1. The document discusses functions of complex variables, including analytic functions, Cauchy-Riemann equations, harmonic functions, and methods for determining an analytic function when its real or imaginary part is known.
2. Some key topics covered are the definition of an analytic function, Cauchy-Riemann equations in Cartesian and polar forms, properties of analytic functions including orthogonal systems, and determining the analytic function using methods like direct, Milne-Thomson's, and exact differential equations.
3. Examples are provided to illustrate determining the analytic function given its real or imaginary part, such as finding the function when the real part is a polynomial or the imaginary part is a trigonometric function.
1. The document contains an assignment on mathematics involving ordinary and partial differential equations, matrices, vector calculus, and their applications. It includes solving various types of differential equations, finding eigenvectors and eigenvalues of matrices, evaluating line, surface and volume integrals using Green's theorem and Stokes' theorem. The assignment contains 20 problems spanning these topics.
2. The assignment covers key concepts in differential equations, linear algebra, and vector calculus including solving ordinary differential equations, partial differential equations, systems of linear equations, eigenproblems, line integrals, surface integrals, divergence, curl, gradient, Laplacian, and theorems like Green's theorem and Stokes' theorem.
3. Students are required to solve 20 problems involving these
This document provides information about a theoretical physics course, including the course code, homepage, lecturer contact information, textbook, main topics covered, and assessment details. The course covers complex variables and their applications in theoretical physics. Topics include Cauchy's integral formula, calculus of residues, partial differential equations, special functions, and Fourier series. Students will be assessed through a 3-hour written exam worth 80% and a course assessment worth 20%.
The document discusses various types of differential equations including ordinary differential equations (ODEs) and partial differential equations (PDEs). It defines key terms like order, degree, and describes several methods for solving common types of differential equations, such as separating variables, exact differentials, linear equations, Bernoulli's equation, and Clairaut's equation. It also includes sample problems and solutions for each method and concludes with multiple choice questions.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IIRai University
This document provides an overview of Unit II - Complex Integration from the Engineering Mathematics-IV course at RAI University, Ahmedabad. It covers key topics such as:
1) Complex line integrals and Cauchy's integral theorem which states that the integral of an analytic function around a closed curve is zero.
2) Cauchy's integral formula which can be used to evaluate integrals and find derivatives of analytic functions.
3) Taylor and Laurent series expansions of functions, including their regions of convergence.
4) The residue theorem which can be used to evaluate real integrals involving trigonometric or rational functions.
The document provides an overview of functions of a complex variable. Some key points:
1) Functions of a complex variable provide powerful tools in theoretical physics for quantities that are complex variables, evaluating integrals, obtaining asymptotic solutions, and performing integral transforms.
2) The Cauchy-Riemann equations are a necessary condition for a function f(z) = u(x,y) + iv(x,y) to be differentiable at a point. If the equations are satisfied, the function is analytic.
3) Cauchy's integral theorem states that if a function f(z) is analytic in a simply connected region R, the contour integral of f(z) around any closed path in
This document discusses complex numbers and functions. It introduces complex numbers using Cartesian (x + iy) and polar (r(cosθ + i sinθ)) forms. It describes the Cauchy-Riemann conditions that must be satisfied for a function of a complex variable to be differentiable. A function is analytic if it satisfies the Cauchy-Riemann conditions and its partial derivatives are continuous. Analytic functions have properties like equality of second-order partial derivatives and establishing a relation between the real and imaginary parts.
1) Complex numbers can be represented in Cartesian (x + iy) or polar (r(cosθ + i sinθ)) form, with conversions between the two.
2) The derivative of a complex function f(z) is defined if the Cauchy-Riemann equations are satisfied.
3) A function is analytic if it is differentiable and its partial derivatives are continuous, implying the Cauchy-Riemann equations always hold. Analytic functions have properties like equality of second partial derivatives.
Contour integrals provide a useful technique for evaluating certain integrals. This document introduces contour integrals and provides examples to illustrate their application. It explains that contour integrals can be viewed as line integrals and that Cauchy's theorem allows continuous deformation of the contour without changing the result, as long as it does not cross singularities. The key points are:
1) Contour integrals around a pole give the residue of the pole times 2πi.
2) Examples are worked out to show contour integrals give the same results as conventional integrals for integrals that cannot be expressed in terms of elementary functions.
3) Contour integrals enable evaluation of new integrals that would be difficult to evaluate otherwise, like calculating an integral of an exponential
This document introduces complex integration and provides examples of evaluating integrals along paths in the complex plane. It expresses integrals in terms of real and imaginary parts involving line integrals of functions. Key points made include:
- Complex integrals can be interpreted as line integrals over paths in the complex plane.
- Integrals of analytic functions over closed paths, like the unit circle, may yield simple results like 2πi or 0.
- Blasius' theorem relates forces and moments on a cylinder in fluid flow to complex integrals around the cylinder boundary.
This document discusses various topics related to integration including indefinite and definite integrals, power rules, properties of integrals, integration by parts, u-substitution, and definitions. Some key points covered are:
- Integration was developed independently by Newton and Leibniz in the late 17th century.
- The indefinite integral finds an antiderivative, while the definite integral evaluates the area under a function between bounds.
- Common integration techniques include power rules, integration by parts, and u-substitution.
- Integration rules and properties allow integrals to be transformed and simplified.
(1) The document discusses various integration techniques including: review of integral formulas, integration by parts, trigonometric integrals involving products of sines and cosines, trigonometric substitutions, and integration of rational functions using partial fractions.
(2) Examples are provided to demonstrate each technique, such as using integration by parts to evaluate integrals of the form ∫udv, using trigonometric identities to reduce powers of trigonometric functions, and using partial fractions to break down rational functions into simpler fractions.
(3) The key techniques discussed are integration by parts, trigonometric substitutions to transform integrals involving quadratic expressions into simpler forms, and partial fractions to decompose rational functions for integration. Various examples illustrate the
Here are the key steps:
1) Choose u and dv based on LIPET:
u = ex
dv = cos x dx
2) Find du and v:
du = ex dx
v = sin x
3) Apply integration by parts formula:
∫uex dx = uv - ∫vdu
= exsinx - ∫sinxexdx
4) Repeat integration by parts on the second term:
∫sinxexdx = excosx - ∫-cosxexdx
5) Combine like terms:
exsinx + excosx - ∫excosxdx
6) The integral on the right is
This document discusses transformations of functions including translations, stretches, reflections, and combinations of functions. It begins by explaining how translating a function's graph vertically or horizontally shifts the graph up/down or left/right by a given amount. Stretching or shrinking a graph vertically stretches or shrinks the y-values, while reflecting graphs flips them over an axis. Functions can also be combined by addition, subtraction, multiplication, composition. Composition applies one function to the output of another. Several examples demonstrate applying transformations to graphs and combining simple functions.
This document provides formulas and methods for solving ordinary differential equations and vector calculus problems that are covered in an Engineering Mathematics course. It includes:
1. Seven methods for finding the complementary function for ODEs with constant coefficients depending on the nature of the roots.
2. Methods for finding the particular integral for ODEs with constant coefficients, including four types of functions the right side could be.
3. An overview of key concepts in vector calculus including vector differential operators, gradient, divergence, curl, and theorems like Green's theorem, Stokes' theorem, and Gauss' divergence theorem.
This document provides an overview of functions of several variables. It discusses notation for functions with multiple independent variables, domains of such functions, and graphs of functions with two or more variables. Specifically, it gives examples of finding the domain of functions defined by equations, sketching the graph of a function as a surface in 3D space using traces in coordinate planes and parallel planes, and creating a contour map using level curves representing different values of the dependent variable.
This document introduces calculus concepts like differentiation and gradients. It discusses:
- Calculating the gradient of a curve at a point from two given points on the curve.
- Using limits to find the gradient of a tangent line as another point on the curve approaches the point of tangency.
- Differentiating polynomials from first principles by considering small changes in x and y.
- Noting that the derivative of x^n is nx^{n-1} and the derivative of ax^n is anx^{n-1}.
- Introducing concepts like the second derivative and using derivatives to find equations of tangents and normals to curves.
The document provides an overview of the finite element method (FEM) formulation. It begins by discretizing the Poisson equation over a domain using basis functions to approximate the solution. Integrating the equation against the basis functions results in a system of equations in matrix form (Ax=b) where the stiffness matrix A and load vector b are defined. For a regular grid, the basis functions and their gradients are used to calculate the elements of A. Boundary conditions are incorporated by specifying function values at nodes on the domain boundaries. The document outlines the basic steps to derive the discrete FEM system of equations from the governing differential equation.
The document provides an overview of the finite element method (FEM) formulation. It begins by discretizing the Poisson equation over a domain using basis functions to approximate the solution. Integrating the equation against the basis functions results in a system of equations in matrix form (Ax=b) where the stiffness matrix A and load vector b are defined. For a regular grid, the basis functions and their gradients are used to calculate the elements of A. Boundary conditions are incorporated by specifying function values at nodes on the domain boundaries. The document outlines the basic formulation of the FEM to solve partial differential equations numerically.
This document contains the solutions to 5 questions related to calculus concepts like integration, derivatives, series approximation, and geometry of curves and surfaces. Some of the key steps include:
- Using integration to find volumes, masses, and centroids
- Finding critical points and classifying extrema
- Approximating a series to evaluate an integral
- Solving a geometric series problem to find an initial height
- Analyzing motion problems using kinematic equations
- Finding equations of planes and tangent lines to surfaces
The document provides an introduction to partial differential equations (PDEs). Some key points:
- PDEs involve functions of two or more independent variables, and arise in physics/engineering problems.
- PDEs contain partial derivatives with respect to two or more independent variables. Examples of common PDEs are given, including the Laplace, wave, and heat equations.
- The order of a PDE is defined as the order of the highest derivative. Methods for solving PDEs through direct integration and using Lagrange's method are briefly outlined.
This document discusses complex functions and their derivatives. It defines a complex function as a function f(z) that maps complex numbers to complex numbers. The derivative of a complex function is defined as the limit of the difference quotient, which may not always exist. Some simple functions like Re(z) are shown to not have complex derivatives. The usual rules of differentiation, such as the sum, product, quotient and chain rules, are shown to hold for complex differentiable functions.
This document provides information about integration in higher mathematics. It begins with an overview of integration as the opposite of differentiation. It then discusses using antidifferentiation to find integrals by reversing the power rule for differentiation. Several examples are provided to illustrate integrating polynomials. The document also discusses using integrals to find the area under a curve or between two curves. It provides examples of calculating areas bounded by graphs and the x-axis. Finally, it presents some exam-style integration questions for practice.
This document discusses convex optimization and properties of convex functions and sets. It begins by defining a convex set as a set where the line segment between any two points in the set is also contained in the set. It then discusses properties of convex sets such as intersections of convex sets being convex and projections onto convex sets being unique. The document defines convex functions and shows examples, and discusses properties such as convex functions being continuous. It also discusses characterizations of convexity using first and second order conditions. Finally, it discusses how convex optimization problems can be solved provably to the global optimum using algorithms like gradient descent.
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UiPath Test Automation using UiPath Test Suite series, part 5DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 5. In this session, we will cover CI/CD with devops.
Topics covered:
CI/CD with in UiPath
End-to-end overview of CI/CD pipeline with Azure devops
Speaker:
Lyndsey Byblow, Test Suite Sales Engineer @ UiPath, Inc.
UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
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1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
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This paper presents Reef, a system for generating publicly verifiable succinct non-interactive zero-knowledge proofs that a committed document matches or does not match a regular expression. We describe applications such as proving the strength of passwords, the provenance of email despite redactions, the validity of oblivious DNS queries, and the existence of mutations in DNA. Reef supports the Perl Compatible Regular Expression syntax, including wildcards, alternation, ranges, capture groups, Kleene star, negations, and lookarounds. Reef introduces a new type of automata, Skipping Alternating Finite Automata (SAFA), that skips irrelevant parts of a document when producing proofs without undermining soundness, and instantiates SAFA with a lookup argument. Our experimental evaluation confirms that Reef can generate proofs for documents with 32M characters; the proofs are small and cheap to verify (under a second).
Paper: https://eprint.iacr.org/2023/1886
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Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
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At WSTS 2024, Alon Stern explored the topic of parametric holdover and explained how recent research findings can be implemented in real-world PNT networks to achieve 100 nanoseconds of accuracy for up to 100 days.
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The choice of an operating system plays a pivotal role in shaping our computing experience. For decades, Microsoft's Windows has dominated the market, offering a familiar and widely adopted platform for personal and professional use. However, as technological advancements continue to push the boundaries of innovation, alternative operating systems have emerged, challenging the status quo and offering users a fresh perspective on computing.
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Maruthi Prithivirajan, Head of ASEAN & IN Solution Architecture, Neo4j
Get an inside look at the latest Neo4j innovations that enable relationship-driven intelligence at scale. Learn more about the newest cloud integrations and product enhancements that make Neo4j an essential choice for developers building apps with interconnected data and generative AI.
3. 3
Text Book: Lecture Notes Selected from
Mathematical Methods for Physicists
International Edition (4th or 5th or 6th Edition)
by
George B. Arfken and Hans J. Weber
Main Contents:
Application of complex variables,
e.g. Cauchy's integral formula, calculus of residues.
Partial differential equations.
Properties of special functions
(e.g. Gamma functions, Bessel functions, etc.).
Fourier Series.
Assessment:
One 3-hour written examination (80% weighting)
and course assessment (20% weighting)
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4. 1 Functions of A Complex Variables I
Functions of a complex variable provide us some powerful and
widely useful tools in in theoretical physics.
• Some important physical quantities are complex variables (the
wave-function Y)
0
• Evaluating definite integrals.
• Obtaining asymptotic solutions of differentials
equations.
• Integral transforms
• Many Physical quantities that were originally real become complex
as simple theory is made more general. The energy
( ® the finite life time).
E ®E + iG n n
1/ G
4Cssfounder.com
5. 1.1 Complex Algebra
We here go through the complex algebra briefly.
A complex number z = (x,y) = x + iy, Where.
We will see that the ordering of two real numbers (x,y) is significant,
i.e. in general x + iy ¹ y + ix
X: the real part, labeled by Re(z); y: the imaginary part, labeled by Im(z)
Three frequently used representations:
(1) Cartesian representation: x+iy
(2) polar representation, we may write
z=r(cos q + i sinq) or
r – the modulus or magnitude of z
q - the argument or phase of z
i = -1
5
z = r ×eiq
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6. r – the modulus or magnitude of z
q - the argument or phase of z
The relation between Cartesian
and polar representation:
r = z = x +
y
q =
- y x
The choice of polar representation or Cartesian representation is a
matter of convenience. Addition and subtraction of complex variables
are easier in the Cartesian representation. Multiplication, division,
powers, roots are easier to handle in polar form,
6
( )
( )
2 2 1/ 2
tan 1 /
z ± z = x ± x + i y ±
y
( ) ( )
1 2 1 2 1 2
z z = x x - y y + i x y +
x y
( ) ( )
1 2 1 2 1 2 1 2 2 2
( 1 2 ) 1 2 1 2
z z =r r eiq +q
( ) ( 1 2 ) 1 / 2 1 / 2 z z = r r ei q -q z n =r neinq
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7. Using the polar form,
arg( ) arg( ) arg( ) 1 2 1 2 z z = z + z
From z, complex functions f(z) may be constructed. They can be written
f(z) = u(x,y) + iv(x,y)
in which v and u are real functions.
For example if , we have
f (z) = z2
The relationship between z and f(z) is best pictured as a mapping
operation, we address it in detail later.
7
z1z2 =z1 z2
f ( z) = (x2 - y2 )+ i2xy
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8. 8
Function: Mapping operation
x
y Z-plane
u
v
The function w(x,y)=u(x,y)+iv(x,y) maps points in the xy plane into points
in the uv plane.
e i
q
= +
i
e = +
i
q q
cos sin
in n
q q
(cos sin )
q
Since
We get a not so obvious formula
cos nq + i sin nq = (cosq + i sinq )n
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9. 9
Complex Conjugation: replacing i by –i, which is denoted by (*),
z* = x -iy
We then have
zz* = x2 + y2 = r 2
z = (zz* )1 2 Special features: single-valued function of a
Hence
Note:
real variable ---- multi-valued function
z = reiq
rei(q +2np )
ln z = ln r + iq
ln z = ln r + i(q + 2np )
ln z is a multi-valued function. To avoid ambiguity, we usually set n=0
and limit the phase to an interval of length of 2p. The value of lnz with
n=0 is called the principal value of lnz.
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10. 10
Another possibility
> ®¥
£
x x
| sin |,| cos | 1 for a real x;
z z
however, possibly | sin |,| cos | 1 and even
Question:
Using the identities :
e - iz e -
iz
= + = -
i
iz iz
2
; sinz e
x iy x y i x y
2
+ = +
to show (a) sin( ) sin cosh cos sinh
x + iy = x y -
i x y
cos( ) cos cosh sin sinh
2 2 2
x y
= +
(b) | sinz | sin sinh
2 2 2
x y
| cosz | cos sinh
cosz e
= +
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11. 1.2 Cauchy – Riemann conditions
11
Having established complex functions, we now proceed to
differentiate them. The derivative of f(z), like that of a real function, is
defined by
df
f z
f z z f z
+ d
- = = = ¢
lim lim
® ® d
provided that the limit is independent of the particular approach to the
point z. For real variable, we require that
Now, with z (or zo) some point in a plane, our requirement that the
limit be independent of the direction of approach is very restrictive.
Consider
( ) ( ) ( ) f ( z)
dz
z
z
z z
d
d
d 0 d 0
( ) ( ) ( o )
¢ = ¢ = ¢
f x f x f x
lim lim
® + ® -
x x x x
o o
dz =dx + idy
df =du +idv
,
u i v
= +
d d
x i y
f
d
z
d d
d
+
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12. 12
Let us take limit by the two different approaches as in the figure. First,
with dy = 0, we let dx0,
u
d
= æ +
i d
v
ö çè
÷ø
f
d
lim lim
® z
® x
x
z d
x d
d
d 0 d 0
+ ¶
i v
¶
x
= ¶
u
¶
x
Assuming the partial derivatives exist. For a second approach, we set
dx = 0 and then let dy 0. This leads to
v
+ ¶
y
i u
=- ¶
y
f
d
z
lim
z ® d
¶
¶
d 0
If we have a derivative, the above two results must be identical. So,
v
=¶
y
u
¶
x
¶
¶
,
=-¶
v
x
¶
u
y
¶
¶
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13. 13
These are the famous Cauchy-Riemann conditions. These Cauchy-
Riemann conditions are necessary for the existence of a derivative, that
is, if exists, the C-R conditions must hold.
f¢(x)
Conversely, if the C-R conditions are satisfied and the partial
derivatives of u(x,y) and v(x,y) are continuous, exists. f¢((zsee )
the proof
in the text book).
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14. 14
Analytic functions
If f(z) is differentiable at and in some small region around ,
we say that f(z) is analytic at
Differentiable: Cauthy-Riemann conditions are satisfied
the partial derivatives of u and v are continuous
Analytic function:
Property 1:
Ñ2u = Ñ2v = 0
Property 2: established a relation between u and v
Example:
Find the analytic functions ( ) ( , ) ( , )
3 2
a u x y = x -
xy
if ( ) ( , ) 3
b v x y e x
w z u x y iv x y
( ) ( , ) = -
y sin
= +
0 z = z
0 z = z
0 z
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15. 1.3 Cauchy’s integral Theorem
We now turn to integration.
in close analogy to the integral of a real function
The contour is divided into n intervals .Let
wiith for j. Then
'0
z0 ® z
0 1 D = - ® j j j- z z z
provided that the limit exists and is
independent of the details of
z
z
choosing the points and ,
j
j
z
where is a point on the curve bewteen
z z
15
¢
å ( ) ò ( )
=
®¥
D =
0
0 1
lim
z
z
n
j
j j
n
f z z f z dz
n®¥
j
and .
j j-
1
The right-hand side of the above equation is called the contour (path)
integral of f(z)
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16. 16
As an alternative, the contour may be defined by
2 2
2
ò ( ) = ò[ ( ) + ( )][ + ]
1 1
1
2 2
udx vdy i vdx udy
with the path C specified. This reduces the complex integral to the
complex sum of real integrals. It’s somewhat analogous to the case of
the vector integral.
An important example
, ,
x y
c x y
z
c z
f z dz u x y iv x y dx idy
= ò [ - ] + ò[ + ]
1 1
2 2
1 1
x y
x y
x y
c c x y
òc
z ndz
where C is a circle of radius r>0 around the origin z=0 in
the direction of counterclockwise.
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17. In polar coordinates, we parameterize
and , and have
n
1 z dz r i n d
i
¹
0 for n -1
{
=
which is independent of r.
Cauchy’s integral theorem
If a function f(z) is analytical (therefore single-valued) [and its partial derivatives are
continuous] through some simply connected region R, for every closed path C in R,
17
z = reiq
dz = ireiqdq
+ p
ò = ò [ ( + ) ]
q q
p p
2
0
1
exp 1
2 2
c
n
1 for n -1
=
ò f ( z)dz = 0
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18. 18
Stokes’s theorem proof
Proof: (under relatively restrictive condition: the partial derivative of u, v
are continuous, which are actually not required but usually
satisfied in physical problems)
ò f ( z ) dz = ò( udx - vdy ) + i ò( vdx + udy
)
c c c
These two line integrals can be converted to surface integrals by
Stokes’s theorem
ò × = òÑ´ ×
c s
A dl A d s
A Ax x Ay y
= + ds =dxdyz
ò( + ) = ò × = òÑ ´ ×
c c s
Using and
We have
Axdx Aydy A dl A d s
ö
æ
A y x
¶
-
¶
A
ò ÷ ÷ø
ç çè
= dxdy
¶
¶
y
x
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19. 19
For the real part, If we let u = Ax, and v = -Ay, then
u
æ
udx vdy v
ö
- = - ¶
+ ¶
¶
ò ÷ ÷ø
( ) dxdy
y
x
ç çè
¶
v
¶
x
¶
=0 [since C-R conditions ]
c ò
= -¶
For the imaginary part, setting u = Ay and v = Ax, we have
u
y
¶
ö
æ
v
vdx udy u
òf (z)dz = 0
+ = ¶ dxdy 0
- ¶
¶
( ) ò ò = ÷ ÷ø
ç çè
¶
y
x
As for a proof without using the continuity condition, see the text book.
The consequence of the theorem is that for analytic functions the line
integral is a function only of its end points, independent of the path of
integration,
1
ò ( ) = ( ) - ( ) = -ò ( )
2
2
1
2 1
z
z
z
z
f z dz F z F z f z dz
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20. 20
•Multiply connected regions
The original statement of our theorem demanded a simply connected
region. This restriction may easily be relaxed by the creation of a
barrier, a contour line. Consider the multiply connected region of
Fig.1.6 In which f(z) is not defined for the interior R¢
1.6 Fig.
Cauchy’s integral theorem is not valid for the contour C, but we can
construct a C¢ for which the theorem holds. If line segments DE and
GA arbitrarily close together, then
E
ò ( ) =-ò ( )
D
A
G
f z dz f z dz
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21. ù
é
ò = ò + ò + ò + ò
¢
f z dz f ( z)dz
'2
é
= ò + ò
ò ( ) ò ( ) ¢ ¢
f z dz f z dz
'1
C
ù
ABD®C EFG®-C
21
( )
( )
ABD DE GA EFG
ABDEFGA
ú ú
û
ê ê
ë
f (z)dz 0
ABD EFG
= úû
êë
=
C1 C2
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22. 1.4 Cauchy’s Integral Formula
22
Cauchy’s integral formula: If f(z) is analytic on and within a closed contour C then
( ) ( 0 )
f z dz
0
2 if z
ò -
z z
C
= p
in which z0 is some point in the interior region bounded by C. Note that
here z-z0 ¹0 and the integral is well defined.
Although f(z) is assumed analytic, the integrand (f(z)/z-z0) is not
analytic at z=z0 unless f(z0)=0. If the contour is deformed as in Fig.1.8
Cauchy’s integral theorem applies.
So we have
ò ( ) ò f ( z
) =
-
-
f z dz
-
C C
dz
z z
z z
2
0
0 0
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23. 23
- = iq
0 z z re
( ) ( ) q q
Let , here r is small and will eventually be made to
approach zero
i
ò ò +
f z dz i
z z
=
-
0
C C
2 2
(r®0)
f z re
i
q
q
rie d
0
re
dz
= ò q = p
( 0 ) 2 ( 0 )
if z d if z
2
C
Here is a remarkable result. The value of an analytic function is given at
an interior point at z=z0 once the values on the boundary C are
specified.
What happens if z0 is exterior to C?
In this case the entire integral is analytic on and within C, so the
integral vanishes.
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24. f z dz
0 0
i 0, z exterior
1
Derivatives
Cauchy’s integral formula may be used to obtain an expression for
the derivation of f(z)
( ) ( )
d f z dz f z
¢ = 0
ç Ñ
ò- ¸ è ø æ ö
1
2
dz p i z z
0 0
æ
-
1
f z dz d
pi p
Moreover, for the n-th order of derivative
24
ò ( ) ( ) =
-
î í ì
C
f z
z z
, z interior
2
0
p 0
ò ( ) ÷ 1 ö
( )
÷ø
= 1
ò ( -
) ç çè
f z dz
= 2
0 0 2 0
2
z z
dz z z i
( ) ( ) ( )
f z dz
ò( - ) +
= 1
0
0 2
!
n
n
z z
i
f z n
p
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25. We now see that, the requirement that f(z) be analytic not only guarantees a first
derivative but derivatives of all orders as well! The derivatives of f(z) are
automatically analytic. Here, it is worth to indicate that the converse of Cauchy’s
integral theorem holds as well
Morera’s theorem:
If a function f(z) is continuous in a simply connected region R
ò f z dz
= C
and ( ) 0 for every closed C within R, then f(z) is
analytic throught R (see the text book).
25
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26. 26
Examples
1. If ( ) a is analytic on and within
0
n
a circle about the origin, find .
n
n
n
a
å³
f z =
z ³ - å
( ) ( ) { } n j
f j z j a a z -
j n
n j
= +
1
!
( ) ( ) j
f j 0 = j!a
( ) ( = 0
) 1
ò ( ) = !
2 + 1
f z dz
n
n
n n i
z
a f
p
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27. 27
f ( z) £ M
2.In the above case, on a circle of radius r about the origin,
then (Cauchy’s inequality)
Proof:
M
a = f z dz £ M r 2
p
r
£ +
ò +1 2 1
1
n n r
where
a r n M
n £
( ) ( ) n n
z r
r
z
=
p
2
p
M(r) =Max z =r f (r)
3. Liouville’s theorem: If f(z) is analytic and bounded in the complex
plane, it is a constant.
Proof: For any z0, construct a circle of radius R around z0,
( ) ( )
M R
f z dz
£
( )2 2
0
0
2
1
2 2
R
z z
i
f z
R
p
p p
-
¢ = ò
=M
R
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28. Since R is arbitrary, let , we have
Conversely, the slightest deviation of an analytic function from a
constant value implies that there must be at least one singularity
somewhere in the infinite complex plane. Apart from the trivial constant
functions, then, singularities are a fact of life, and we must learn to live
with them, and to use them further.
28
R ® ¥
f ¢(z) =0, i.e, f (z) =const.
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29. 1.5 Laurent Expansion
29
Taylor Expansion
Suppose we are trying to expand f(z) about z=z0, i.e.,
and we have z=z1 as the nearest point for which f(z) is not analytic. We
construct a circle C centered at z=z0 with radius
From the Cauchy integral formula,
( ) ( ) å¥
= -
n -
0
n
n 0 f z a z z
z¢ - z0 < z1 - z0
( ) ( ) ( )
¢ ¢
¢ ¢
f z dz
1
f z dz
ò =
¢ -
p
ò ( ¢ - ) - ( - )
f z 1
p
=
z z
2 i
z z z z
C C 0 0 2 i
( )
¢ ¢
f z dz
ò ( ¢ - )[ - ( - ) ( ¢ - )]
1
p
=
C 0 0 0 z z 1 z z z z
2 i
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30. 30
Here z¢ is a point on C and z is any point interior to C. For |t| <1, we
note the identity
t t t n
1 2
= + + + =
1
t
- 0
1
So we may write
å¥
=
n
( ) ( ) n
( )
- ¢ ¢
z z f z dz
( ) òå¥
=
n
0
¢- +
=
1
p
C n
z z
i
f z
0
1
0
2
å ¥
( ) ò( ( )
)
1
z z f z dz
= -
2 0
n
pi
( )
n =
C
¢ ¢
n
¢- +
0
1
0
z z
( ) ( ) å¥
=
= -
n
n
f z
which is our desired 0
Taylor expansion, just as for real variable power
series, this expansion is unique for a given z0.
0
0 n
! n
z z
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31. 31
Schwarz reflection principle
From the binomial expansion of g ( z ) = ( for z - integer x0
)n
n (as an
assignment), it is easy to see, for real x0
( ) [( ) ] ( )n ( * )
g* z = z - x = z - x = g z
0
n * *
0
Schwarz reflection principle:
If a function f(z) is (1) analytic over some region including the real axis
and (2) real when z is real, then
f * ( z) = f (z* )
We expand f(z) about some point (nonsingular) point x0 on the real axis
because f(z) is analytic at z=x0.
( ) ( ) å¥ =
( ) ( )
f z = z -
x
n
n
f x
0 ! n
0
0
n
Since f(z) is real when z is real, the n-th derivate must be real.
( ) å¥
=( - ) n ( n ) ( ) = ( *
) 0
0
0
* *
f x
!
f z
n
f z z x
n
=
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32. 32
Laurent Series
We frequently encounter functions that are analytic in annular region
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33. Drawing an imaginary contour line to convert our region into a simply
connected region, we apply Cauchy’s integral formula for C2 and C1,
with radii r2 and r1, and obtain
ù
é
( ) ( )
¢ ¢
f z dz
z z
1
p
= ò - ò
We let r2 ®r and r1 ®R, so for C1, while for C2, .
We expand two denominators as we did before
33
¢ ¢
f z dz
= ò ò
f z dz
2 0 1 0 0 0 1 0 0
z z f z dz
¥
1
å¥
1
1
p p
( ) ( ) n ¢ - ¢ ¢
1
2
(Laurent Series)
i
f z
C C
¢-
ú ú ú
û
ê ê ê
ë
1 2
2
0 0 z¢ - z > z - z
0 0 z¢ - z < z - z
ì
( ) ( )
( )[ ( ) ( )]
( )
ü
ïý
¢ ¢
( ) ( ) ( ) [ ]ïþ
ïî
ïí
- - ¢ - -
+
¢ - - - ¢ -
1 2
1
C C
z z z z z z
z z z z z z
i
f z
p
( ) ( )
( ) ( )
( z z ) f ( z )dz
z z i z z
i
n
n
n C
n C
n
-
+
¢ -
¢ ¢
= å - ò å ò
=
+
¥
=
+
0
1 0
0 0
0
0
1 2
2
= -
n
=-¥
n
f z an z z0
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34. 34
where
ò( ( ¢ )
¢
¢- ) +
1
p
a =
1
f z dz
n n z z
0 2
C
i
Here C may be any contour with the annular region r < |z-z0| < R
encircling z0 once in a counterclockwise sense.
Laurent Series need not to come from evaluation of contour integrals.
Other techniques such as ordinary series expansion may provide the
coefficients.
Numerous examples of Laurent series appear in the next chapter.
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35. Example:
(1) Find Taylor expansion ln(1+z) at point z
(2) find Laurent series of the function
z z
+ = - +
ln(1 ) ( 1) 1
- ¢ ¢ =
¢
z dz
dz
( ) ( ) ò òå¥
+ ¢ +
1 2 2
¥
i
rie d
åò ( )
1
If we employ the polar form
a = -
+ - + - q
n r e
2 2 2
=
0
1
m
n m i n m
i
q q
p
=
¢ ¢ -
¢
=
0
1
1
1
2
m
n
m
n i z
n z z i
z
a
p p
f (z) [z(z 1)] 1 = - -
35
å¥
1
i n m
=- × + -
2 2 ,1
=
0
2
m
i
p d
p
- ³
î í ì
1 for n -1
<
=
0 for n -1
a n
1 1 2 3
( ) å¥
z z z zn
= -
= - - - - - - = -
- 1
1
1
n
z z z
The Laurent expansion becomes
å¥
=
1
n
n
n
n
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36. Analytic continuation
For example
which has a simple pole at z = -1 and is analytic
elsewhere. For |z| < 1, the geometric series expansion f1, while expanding
it about z=i leads to f2,
; ( ) ; 1
å å¥
=
¥
1 1
=
z i
- -
ö çè
÷ø
æ
+
+
= - =
( ) 1
+
=
0
2
0
1
n
n
n
n
z i
i
f z f
z
f z
36
f (z) =1 (1+ z)
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37. 37
( ) å¥
=
z z z n
1 2
= - + + = -
1
z
+ 0
1
n
Suppose we expand it about z = i, so that
1
f z 1
( ) =
1 + i + ( z -
i
) =
( 1 + i )[ 1 + ( z - i ) ( 1 +
i
)] ( )
( ) úû
é
2
z i
1 z i
1
= +
2
+ -
- -
converges for (Fig.1.10)
ù
êë
+
+
+
1 i
1 i
1 i
z - i < 1+ i = 2
The above three equations are different representations of the same
function. Each representation has its own domain of convergence.
A beautiful theory:
If two analytic functions coincide in any region, such as the overlap of s1 and s2,
of coincide on any line segment, they are the same function in the sense that they
Cwisll csofioncuidne deveerry.wchoerme as long as they are well-defined.