Complex Variables and Applications - student soluions manual

Preface

Chapter 1  Complex Numbers
     1 Sums and Products
     2 Basic Algebraic Properties
     3 Further Properties
     4 Moduli
     5 Complex Conjugates
     6 Exponential Fonn
     7 Products and Quotients in Exponential Form
     8 Roots of Complex Numbers
     9 Examples
     10 Regions in the Complex Plane

Chapter 2  Analytic Functions
     11 Functions of a Complex Variable
     12 Mappings
     13 Mappings by the Exponential Function
     14 Limits
     15 Theorems on Limits
     16 Limits Involving the Point at Infinity
     17 Continuity
     18 Derivatives
     19 Differentiation Formulas
     20 Cauchy-Riemann Equations
     21 Sufficient Conditions for Differentiability
     22 Polar Coordinates
     23 Analytic Functions
     24 Examples
     25 Harmonic Functions
     26 Uniquely Determined Analytic Functions
     27 Reflection Principle

Chapter 3  Elementary Functions
     28 The Exponential Function
     29 The Logarithmic Function
     30 Branches and Derivatives of Logarithms
     31 Some Identities Involving Logarithms
     32 Complex Exponents
     33 Trigonometric Functions
     34 Hyperbolic Functions
     35 Inverse Trigonometric and Hyperbolic Functions

Chapter 4  Integrals
     36 Derivatives of Functions w(t)
     37 Definite Integrals of Functions w(t)
     38 Contours
     39 Contour Integrals
     40 Examples
     41 Upper Bounds for Moduli of Contour Integrals
     42 Antiderivatives
     43 Examples
     44 Cauchy-Goursat Theorem
     45 Proof of the Theorem
     46 Simply and Multiply Connected Domains
     47 Cauchy Integral Formula
     48 Derivatives of Analytic Functions
     49 Liouville\'s Theorem and the Fundamental Theorem of Algebra
     50 Maximum Modulus Principle

Chapter 5  Series
     51 Convergence of Sequences
     52 Convergence of Series
     53 Taylor Series
     54 Examples
     55 Laurent Series
     56 Examples
     57 Absolute and Uniform Convergence of Power Series
     58 Continuity of Sums of Power Series
     59 Integration and Differentiation of Power Series
     60 Uniqueness of Series Representations
     61 Multiplication and Division of Power Series

Chapter 6  Residues and Poles
     62 Residues
     63 Cauchy\'s Residue Theorem
     64 Using a Single Residue
     65 The Three Types of Isolated Singular Points
     66 Residues at Poles
     67 Examples
     68 Zeros of Analytic Functions
     69 Zeros and Poles
     70 Behavior off Near Isolated Singular Points

Chapter 7  Applications of Residues
     71 Evaluation of Improper Integrals
     72 Example
     73 Improper Integrals from Fourier Analysis
     74 Jordan\'s Lemma
     75 Indented Paths
     76 An Indentation Around a Branch Point
     77 Integration Along a Branch Cut
     78 Definite Integrals involving Sines and Cosines
     79 Argument Principle
     80 Roucht\'s Theorem
     81 Inverse Laplace Transforms
     82 Examples

Chapter 8  Mapping by Elementary Functions
     83 Linear Transformations
     84 The Transformation w = l/z
     85 Mappings by 1/z
     86 Linear Fractional Transformations
     87 An Implicit Fonn
     88 Mappings of the Upper Half Plane
     89 The Transformation w = sin z
     90 Mappings by z^2 and Branches of z^{1/2}
     91 Square Roots of Polynomials
     92 Riemann Surfaces
     93 Surfaces for Related Functions

Chapter 9  Conformal Mapping
     94 Preservation of Angles
     95 Scale Factors
     96 Local Inverses
     97 Harmonic Conjugates
     98 Transformations of Harmonic Functions
     99 Transformations of Boundary Conditions

Chapter 10 Applications of Conformal Mapping
     100 Steady Temperatures
     101 Steady Temperatures in a Half Plane
     102 A Related Problem
     103 Temperatures in a Quadrant
     104 Electrostatic Potential
     105 Potential in a Cylindrical Space
     106 Two-Dimensional Fluid How
     107 The Stream Function
     108 Flows Around a Comer and Around a Cylinder

Chapter 11 The Schwarz-Christoffel Transformation
     109 Mapping the Real Axis onto a Polygon
     110 Schwarz-Christoffel Transformation
     111 Triangles and Rectangles
     112 Degenerate Polygons
     113 Fluid Flow in a Channel Through a Slit
     114 Flow in a Channel with an Offset
     115 Electrostatic Potential about an Edge of a Conducting Plate

Chapter 12 Integral Formulas of the Poisson Type
     116 Poisson Integral Formula
     117 Dirichlet Problem for a Disk
     118 Related Boundary Value Problems
     119 Schwarz Integral Formula
     120 Dirichlet Problem for a Half Plane
     121 Neumann Problems

Appendix 1  Bibliography

Appendix 2  Table of Transformations of Regions

Index