Complex Variables: Principles And Problem Sessions

CONTENTS
Preface
Acknowledgments
To the Reader
Notation and Symbols
Principles
1. Complex Numbers
§1.1 Introduction
§1.1.1 Complex conjugate
§1.1.2 Polar form
§1.1.3 Shifted polar form
§1.1.4 Representation of angles
§1.2 Examples
§1.3 Linear Transformations
§1.3.1 Complex numbers as vectors in a plane
§1.3.2 Translation, scaling, and rotation
§1.4 Reflection and Inversion
§1.5 Examples
§1.6 Further Topics
§1.6.1 Point at infinity
§1.6.2 Stereographic projection
§1.6.3 Bilinear transformations
§1.7 Notes and References
2. Elementary Functions and Differentiation
§2.1 Exponential, Trigonometric, and Hyperbolic Functions
§2.2 Solutions to Equations
§2.3 Examples
§2.4 Open Sets, Domains, and Regions
§2.5 Limit, Continuity, and Differentiation
§2.6 Cauchy–Riemann Equations
§2.6.1 Cauchy–Riemann equations in polar form
§2.6.2 Sufficiency conditions
§2.7 Examples
§2.8 Analytic Functions
§2.8.1 Properties of analytic functions
§2.8.2 Power series as an analytic function
§2.9 Harmonic functions
§2.10 Graphical Representation of Functions
§2.11 Notes and References
3. Functions with Branch Point Singularity
§3.1 Inverse Functions
§3.1.1 Many-to-one functions
§3.1.2 Inverse functions
§3.1.3 Branch of a multivalued function
§3.2 Nature of Branch Point Singularity
§3.2.1 Multivalued function arg(z)
§3.2.2 Single-valued branches of the square root and the cube root
§3.3 Ensuring Single-Valuedness
§3.3.1 Branch cut
§3.4 Defining a Single-Valued Branch
§3.4.1 Principal value of the logarithm and power
§3.4.2 Examples
§3.5 Multivalued Functions of z . ξ
§3.5.1 Sum and product of √z + 1 and √z .1
§3.6 Discontinuity Across the Branch Cut
§3.7 Examples
§3.8 Inverse Trigonometric Functions
§3.9 Differentiation
§3.10 Riemann Surface
§3.11 Summary
§3.12 Notes and References
4. Integration in the Complex Plane
§4.1 Improper Integrals
§4.2 Definitions
§4.2.1 Integration in the complex plane
§4.3 Examples of Line Integrals in the Complex Plane
§4.4 Bounds on Integrals
§4.4.1 Jordan’s lemma
§4.5 Examples
§4.6 Cauchy’s Fundamental Theorem
§4.6.1 Cauchy’s theorem
§4.6.2 Deformation of contours
§4.6.3 Indefinite integral
§4.7 Transforming Integrals over a Real Interval into Contour Integrals
§4.7.1 Adding line segments or circular arcs to close the contour
§4.7.2 Translation and rotation of the contour
§4.8 Integration of Multivalued Functions
§4.8.1 Line integrals
§4.8.2 Indefinite integrals
§4.8.3 Case study of the indefinite integral dz z
§4.8.4 Integration around a branch cut
§4.9 Summary
5. Cauchy’s Integral Formula
§5.1 Cauchy’s Integral Formula
§5.2 Existence of Higher Order Derivatives
§5.3 Taylor Series
§5.4 Real Variable vs. Complex Variable
§5.5 Examples
§5.6 Laurent Expansion
§5.7 Examples
§5.8 Taylor and Laurent Series for Multivalued Functions
§5.9 More Results Flowing from the Integral Formula
§5.10 Analytic Continuation
§5.10.1 Analytic continuation
§5.10.2 Uniqueness of analytic continuation
§5.10.3 Analytic function as a single entity
§5.10.4 Schwarz reflection principle
§5.10.5 An application
§5.11 Summary
6. Residue Theorem
§6.1 Classification of Singular Points
§6.1.1 Behavior near an isolated singular point
§6.2 Finding the Order of Poles and Residues
§6.3 Residue at an Isolated Singular Point
§6.3.1 Residue at a pole
§6.4 Computing the Residues
§6.5 Cauchy’s Residue Theorem
§6.5.1 An integral for indented contours.
§6.6 Residue at Infinity
§6.7 Illustrative Examples
§6.8 Residue Calculus and Multivalued Functions
§6.9 Zeros and Poles of a Meromorphic Function
§6.10 Notes and References
7. Contour Integration
§7.1 Rational and Trigonometric Functions
§7.2 Integration Around a Branch Cut
§7.3 Using Indented Contours for Improper Integrals
§7.3.1 An example
§7.4 Indented Contours for Singular Integrals
§7.4.1 Definition using indented contours
§7.4.2 Using the i prescription
§7.4.3 Cauchy principal value
§7.5 Miscellaneous Contour Integrals
§7.6 Series Summation and Expansion
§7.6.1 Summation of series
§7.6.2 Mittag–Leffler expansion
§7.7 A Summary
8. Asymptotic Expansion
§8.1 Properties of Asymptotic Expansions
§8.2 Integration by Parts
§8.3 Laplace’s Method
§8.3.1 Dominant term of asymptotic expansion
§8.3.2 Full asymptotic expansion
§8.4 Method of Stationary Phase
§8.5 Method of Steepest Descent
§8.5.1 Central idea
§8.5.2 Local properties of steepest paths
§8.5.3 Change of variable
§8.6 Saddle Point Method
§8.7 Examples
§8.8 Topics for Further Study
9. Conformal Mappings
§9.1 Conformal Mappings
§9.2 Bilinear Transformations
§9.3 Examples
§9.4 Mapping by Elementary Functions
§9.4.1 Mapping w = zn
§9.4.2 Exponential map
§9.4.3 Map w = Log z
§9.4.4 Map w = sin z
§9.5 Joukowski Map
§9.6 Examples
§9.7 Schwarz–Christoffel Transformation
§9.8 Examples
§9.9 Notes and References
10. Physical Applications of Conformal Mappings
§10.1 Model Problems
§10.2 Physical Applications
§10.3 Steady State Temperature Distribution
§10.4 Electrostatic Potential
§10.5 Flow of Fluids
§10.5.1 Stream function and stream lines
§10.6 Solutions Described by Simple Complex Potentials
§10.7 Using Conformal Mappings
§10.8 Method of Images
§10.9 Using the Schwarz–Christoffel Transformation
§10.10 Notes and References
Problem Sessions
1. Complex Numbers
§§1.1 Exercise: Polar Form of Complex Numbers
§§1.2 Exercise: Curves in the Complex Plane
§§1.3 Exercise: Complex Numbers and Geometry
§§1.4 Tutorial: Geometric Representation
§§1.5 Quiz: Transformations
§§1.6 Exercise: Linear Transformations
§§1.7 Exercise: Reflections
§§1.8 Mined: Complex numbers
§§1.9 Mixed Bag: Complex Numbers and Transformations
2. Elementary Functions and Differentiation
§§2.1 Exercise: De Moivre’s Theorem
§§2.2 Exercise: Real and Imaginary Parts
§§2.3 Questions: Hyperbolic and Trigonometric Functions
§§2.4 Exercise: Solutions to Equations
§§2.5 Mined: Solutions to Equations
§§2.6 Tutorial: Roots of a Complex Number
§§2.7 Quiz: Roots of Unity
§§2.8 Exercise: Continuity and Differentiation
§§2.9 Questions: Cauchy–Riemann Equations
§§2.10 Quiz: Cauchy–Riemann Equations
§§2.11 Tutorial: Analytic Functions
§§2.12 Exercise: Harmonic Functions
§§2.13 Mixed Bag: Differentiation and Analyticity
3. Functions with Branch Point Singularity
§§3.1 Questions: Branch Point
§§3.2 Tutorial: Square Root Branch Cut
§§3.3 Exercise: Branch Cut for z-a
§§3.4 Quiz: Discontinuity and Branch Cut
§§3.5 Exercise: Logarithmic Function
§§3.6 Exercise: Discontinuity Across the Branch Cut
§§3.7 Mined: Branch Point Singularity
§§3.8 Mixed Bag: Multivalued Functions
4. Integration in the Complex Plane
§§4.1 Questions: Range of Parameters in Improper Integrals
§§4.2 Tutorial: Computing Line Integrals in the Complex plane
§§4.3 Exercise: Evaluation of Line Integrals
§§4.4 Questions: Deformation of Contours
§§4.5 Exercise: Deformation of Contours
§§4.6 Exercise: Cauchy’s Theorem
§§4.7 Tutorial: Shift of a Real Integration Variable by a Complex Number
§§4.8 Tutorial: Scaling of a Real Integration Variable by a Complex Number
§§4.9 Exercise: Shift and Scaling by a Complex Number
§§4.10 Exercise: Rotation of the Contour
§§4.11 Mixed Bag: Integration in the Complex Plane
5. Cauchy’s Integral Formula
§§5.1 Exercise: Cauchy’s-Integral Formula
§§5.2 Quiz: Circle of Convergence of Taylor Expansion
§§5.3 Exercise: Using MacLaurin’s Theorem
§§5.4 Exercise: Taylor Series Representation
§§5.5 Tutorial: Series Expansion from the Binomial Theorem
§§5.6 Exercise: Laurent Expansion using the Binomial Theorem
§§5.7 Quiz: Subsets for Convergence of Laurent Expansions
§§5.8 Exercise: Laurent Expansion Near a Singular Point
§§5.9 Quiz: Regions of Convergence; Taylor and Laurent Series
§§5.10 Questions: Region of Convergence for Laurent Expansion
6. Residue Theorem
§§6.1 Questions: Classifying Singular Points
§§6.2 Tutorial: Isolated Singular Points
§§6.3 Questions: Selecting Functions with Singularities Specified
§§6.4 Tutorial: Residues at Simple Poles
§§6.5 Tutorial: Residues at Multiple Poles
§§6.6 Exercise: Computation of Integrals
§§6.7 Questions: Residue Theorem
§§6.8 Tutorial: Integrals of Trigonometric Functions
§§6.9 Exercise: Integrals of the Type 2π f(cos θ, sin θ)dθ
§§6.10 Exercise: Integrals Using the Residue at Infinity
§§6.11 Quiz: Finding Residues
§§6.12 Mined: How to Compute Residues
§§6.13 Mixed Bag: Residues and Integration in the Complex Plane
7. Contour Integration
§§7.1 Tutorial: ∞ 0 Q(x) dx
§§7.2 Tutorial: Improper Integrals of Rational Functions
§§7.3 Exercise: Integrals of Type Q(x)dx
§§7.4 Exercise: Integrals of sin x with Rational Functions
§§7.5 Tutorial: Integration Around the Branch Cut
§§7.6 Exercise: Integrals of the Type xaQ(x)dx
§§7.7 Exercise: Integrals of the Type log xQ(x)dx
§§7.8 Tutorial: Hyperbolic Functions
§§7.9 Exercise: Integrals Involving Hyperbolic Functions
§§7.10 Tutorial: Principal Value Integrals
§§7.11 Exercise: Integrals Requiring the Use of Indented Contours
§§7.12 Exercise: Series Summation and Expansion
§§7.13 Exercise: What You See Is Not What You Get
§§7.14 Exercise: Integrals from Statistical Mechanics
§§7.15 Exercise: Alternate Routes to Improper Integrals
§§7.16 Open-Ended: Killing Two Birds with One Stone
§§7.17 Open-Ended: Food for Your Thought
§§7.18 Mixed Bag: Improper Integrals
8. Asymptotic Expansions
§§8.1 Exercise: Integration by Parts
§§8.2 Exercise: Dominant Term
§§8.3 Exercise: Laplace’s Method
§§8.4 Exercise: Steepest Paths
§§8.5 Tutorial: Method of Steepest Descent
§§8.6 Tutorial: Saddle Point Method
§§8.7 Exercise: Steepest Descent and Saddle Point Method
9. Conformal Mapping
§§9.1 Tutorial: Inversion Map
§§9.2 Exercise: Map 1+z 1-z
§§9.3 Exercise: Bilinear Transformation I
§§9.4 Questions: Bilinear Transformation II
§§9.5 Exercise: Symmetry Principle
§§9.6 Exercise: Elementary Functions
§§9.7 Exercise: Finding Maps
§§9.8 Quiz: Matching Aerofoils
§§9.9 Tutorial: Schwarz–Christoffel Transformation
§§9.10 Quiz: Schwarz–Christoffel Mapping
§§9.11 Exercise: Schwarz–Christoffel Mapping
§§9.12 Mixed Bag: Conformal Mappings
10. Physical Applications of Conformal Mappings
§§10.1 Tutorial: Temperature Distribution
§§10.2 Exercise: Steady State Temperature
§§10.3 Exercise: Electrostatics
§§10.4 Quiz: Nine Problems and
§§10.5 Exercise: Flow of Fluids
§§10.6 Exercise: Method of Images
§§10.7 Open-Ended: Using Ideas From Gauss’ Law
§§10.8 Tutorial: Boundary Value Problems
§§10.9 Exercise: Using Schwarz–Christoffel Mapping
§§10.10 Exercise: Utilizing Conservation of Flux of Fluids
§§10.11Mixed Bag: Boundary Value Problems
Bibliography
Index