COMPLEX VARIABLES. Principles and Problem Sessions YJ? A K KAPOOR. University of Hyderabad, India. World Scientific NEW JERSEY LONDON

Transcription

1 COMPLEX VARIABLES Principles and Problem Sessions A K KAPOOR University of Hyderabad, India NEW JERSEY LONDON YJ? World Scientific SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI

2 CONTENTS Preface vii Acknowledgments ix To the Reader xi Notation and Symbols xv Principles 1 1 Complex Numbers 3 11 Introduction Complex conjugate Polar form Shifted polar form Representation of angles 5 12 Examples 7 13 Linear Transformations Complex numbers as vectors in a plane Translation, scaling, and rotation Reflection and Inversion Examples Further Topics Point at infinity Stereographic projection Bilinear transformations Notes and References 23 xvii

3 xviii Complex Variables: Principles and Problem Sessions 2 Elementary Functions and Differentiation Exponential, Trigonometric, and Hyperbolic Functions Solutions to Equations Examples Open Sets, Domains, and Regions Limit, Continuity, and Differentiation Cauchy-Riernann Equations Cauchy-Riemann equations in polar form Sufficiency conditions Examples Analytic Functions Properties of analytic functions Power series as an analytic function Harmonic functions Graphical Representation of Functions Notes and References 49 3 Functions with Branch Point Singularity Inverse Functions Many-to-one functions Inverse functions Branch of a multivalued function Nature of Branch Point Singularity Multivalued function arg(z) Single-valued branches of the square root and the cube root Ensuring Single-Valuedness Branch cut Defining a Single-Valued Branch Principal value of the logarithm and power Examples Multivalued Functions of z Sum and product of \fz + 1 and \/z Discontinuity Across the Branch Cut Examples Inverse Trigonometric Functions Differentiation Riemann Surface Summary Notes and References 76

4 Contents xix 4 Integration in the Complex Plane Improper Integrals Definitions Integration in the complex plane Examples of Line Integrals in the Complex Plane Bounds on Integrals Jordan's lemma Examples Cauchy's Fundamental Theorem Cauchy's theorem Deformation of contours Indefinite integral Transforming Integrals over a Real Interval into Contour Integrals Adding line segments or circular arcs to close the contour Translation and rotation of the contour Integration of Multivalued Functions Line integrals Indefinite integrals Case study of the indefinite integral / y Integration around a branch cut Summary Cauchy's Integral Formula Ill 51 Cauchy's Integral Formula Ill 52 Existence of Higher Order Derivatives Taylor Series Real Variable vs Complex Variable Examples Laurent Expansion Examples Taylor and Laurent Series for Multivalued Functions More Results Flowing from the Integral Formula Analytic Continuation Analytic continuation Uniqueness of analytic continuation Analytic function as a single entity Schwarz reflection principle An application Summary 138

5 xx Complex Variables: Principles and Problem Sessions 6 Residue Theorem Classification of Singular Points Behavior near an isolated singular point Finding the Order of Poles and Residues Residue at an Isolated Singular Point Residue at a pole Computing the Residues Cauchy's Residue Theorem An integral for indented contours Residue at Infinity Illustrative Examples Residue Calculus and Multivalued Functions Zeros and Poles of a Meromorphic Function Notes and References Contour Integration Rational and Trigonometric Functions Integration Around a Branch Cut Using Indented Contours for Improper Integrals An example Indented Contours for Singular Integrals Definition using indented contours Using the ie prescription Cauchy principal value Miscellaneous Contour Integrals Series Summation and Expansion Summation of series Mittag-Leffler expansion A Summary Asymptotic Expansion Properties of Asymptotic Expansions Integration by Parts Laplace's Method Dominant term of asymptotic expansion Full asymptotic expansion Method of Stationary Phase Method of Steepest Descent Central idea Local properties of steepest paths Change of variable 206

6 Contents xxi 86 Saddle Point Method Examples Topics for Further Study Conformal Mappings Conformal Mappings Bilinear Transformations Examples Mapping by Elementary Functions Mapping w zn = Exponential map Map io Log = z Map io sin2; = Joukowski Map Examples Schwarz-Christoffel Transformation Examples Notes and References Physical Applications of Conformal Mappings Model Problems Physical Applications Steady State Temperature Distribution Electrostatic Potential Flow of Fluids Stream function and stream lines Solutions Described by Simple Complex Potentials Using Conformal Mappings Method of Images Using the Schwarz-Christoffel Transformation Notes and References 269 Problem Sessions Complex Numbers Exercise: Polar Form of Complex Numbers Exercise: Curves in the Complex Plane Exercise: Complex Numbers and Geometry Tutorial: Geometric Representation Quiz: Transformations Exercise: Linear Transformations Exercise: Reflections 286

7 xxii Complex Variables: Principles and Problem Sessions 18 Mined: Complex numbers Mixed Bag: Complex Numbers and Transformations Elementary Functions and Differentiation Exercise: De Moivre's Theorem Exercise: Real and Imaginary Parts Questions: Hyperbolic and Trigonometric Functions Exercise: Solutions to Equations Mined: Solutions to Equations Tutorial: Roots of a Complex Number Quiz: Roots of Unity Exercise: Continuity and Differentiation Questions: Cauchy-Riemann Equations Quiz: Cauchy-Riemann Equations Tutorial: Analytic Functions Exercise: Harmonic Functions Mixed Bag: Differentiation and Analyticity Functions with Branch Point Singularity Questions: Branch Point Tutorial: Square Root Branch Cut Exercise: Branch Cut for JO y Z-\ Quiz: Discontinuity and Branch Cut Exercise: Logarithmic Function Exercise: Discontinuity Across the Branch Cut Mined: Branch Point Singularity Mixed Bag: Multivalued Functions Integration in the Complex Plane Questions: Range of Parameters in Improper Integrals Tutorial: Computing Line Integrals in the Complex plane Exercise: Evaluation of Line Integrals Questions: Deformation of Contours Exercise: Deformation of Contours Exercise: Cauchy's Theorem Tutorial: Shift of a Real Integration Variable by a Complex Number Tutorial: Scaling of a Real Integration Variable by a Complex Number Exercise: Shift and Scaling by a Complex Number 350

8 Contents xxiii 410 Exercise: Rotation of the Contour Mixed Bag: Integration in the Complex Plane Cauchy's Integral Formula Exercise: Cauchy's-Integral Formula Quiz: Circle of Convergence of Taylor Expansion Exercise: Using MacLaurin's Theorem Exercise: Taylor Series Representation Tutorial: Series Expansion from the Binomial Theorem 56 Exercise: Laurent Expansion using the Binomial Theorem Quiz: Subsets for Convergence of Laurent Expansions Exercise: Laurent Expansion Near a Singular Point Quiz: Regions of Convergence; Taylor and Laurent Series 510 Questions: Region of Convergence for Laurent Expansion Residue Theorem Questions: Classifying Singular Points Tutorial: Isolated Singular Points Questions: Selecting Functions with Singularities Specified Tutorial: Residues at Simple Poles Tutorial: Residues at Multiple Poles Exercise: Computation of Integrals Questions: Residue Theorem Tutorial: Integrals of Trigonometric Functions Exercise: Integrals of the Type f** /(cos d, sin 8)dQ Exercise: Integrals Using the Residue at Infinity Quiz: Finding Residues Mined: How to Compute Residues Mixed Bag: Residues and Integration in the Complex Plane Contour Integration Tutorial: fq(x)dx Tutorial: Improper Integrals of Rational Functions Exercise: Integrals of Type JQ(x)dx Exercise: Integrals of with Rational Functions 401 \COS ie/ 75 Tutorial: Integration Around the Branch Cut Exercise: Integrals of the Type J xaq{x)dx Exercise: Integrals of the Type JlogxQ(x)dx Tutorial: Hyperbolic Functions Exercise: Integrals Involving Hyperbolic Functions Tutorial: Principal Value Integrals 413

9 xxiv Complex Variables: Principles and Problem Sessions 711 Exercise: Integrals Requiring the Use of Indented Contours Exercise: Series Summation and Expansion Exercise: What You See Is Not What You Get Exercise: Integrals from Statistical Mechanics Exercise: Alternate Routes to Improper Integrals Open-Ended: Killing Two Birds with One Stone Open-Ended: Food for Your Thought Mixed Bag: Improper Integrals Asymptotic Expansions Exercise: Integration by Parts Exercise: Dominant Term Exercise: Laplace's Method Exercise: Steepest Paths Tutorial: Method of Steepest Descent Tutorial: Saddle Point Method Exercise: Steepest Descent and Saddle Point Method Conformal Mapping Tutorial: Inversion Map Exercise: Map Exercise: Bilinear Transformation I Questions: Bilinear Transformation II Exercise: Symmetry Principle Exercise: Elementary Functions Exercise: Finding Maps Quiz: Matching Aerofoils Tutorial: Schwarz-Christoffel Transformation Quiz: Schwarz-Christoffel Mapping Exercise: Schwarz-Christoffel Mapping Mixed Bag: Conformal Mappings Physical Applications of Conformal Mappings Tutorial: Temperature Distribution Exercise: Steady State Temperature Exercise: Electrostatics Quiz: Nine Problems and Exercise: Flow of Fluids Exercise: Method of Images Open-Ended: Using Ideas From Gauss' Law Tutorial: Boundary Value Problems 482

10 Contents xxv 109 Exercise: Using Schwarz-Christoffel Mapping Exercise: Utilizing Conservation of Flux of Fluids Mixed Bag: Boundary Value Problems 489 Bibliography 491 Index 493