Complex Variables I

Solutions to Exercises for Complex Variables by M. Ya. Antimirov, A. A. Kolyshkin and Remi Vaillancourt

Chapter 1: Functions of a Complex Variable

Complex numbers

Continuity in the complex plane

Functions of a complex variable

Analytic functions

Elementary analytic functions

Chapter 2: Elementary Conformal Mappings

Geometric meaning of f'(z)

Basic problems and principles of conformal mappings

Linear mapping and inversion

Linear fractional transformations

Symmetry and linear fractional transformations

Mapping by z^n and w = z^(1/n)

Exponential and logarithmic mappings

Mapping by Joukowsky's function

Mapping by trigonometric functions

Chapter 3: Complex Integration and Cauchy's Theorem

Paths in the complex plane

Complex line integrals

Cauchy's Theorem

Cauchy's integral formula and applications

Goursat's Theorem

Chapter 4: Taylor and Laurent Series

Infinite series

Integer power series

Taylor series

Laurent series

Chapter 5: Singular Points and the Residue Theorem

Singular points of analytic functions

The residue theorem

Chapter 6: Elementary Definite Integrals

Rational functions over (-Infinity,+Infinity)

Rational functions times sine or cosine

Rational functions times exponential functions

Rational functions times a power of x

Chapter 7: Intermediate Definite Integrals

Rational functions over (0, +Infinity)

Forms containing (ln x)^p in the numerator

Forms containing In g(x) or arctan g(x)

Forms containing In in the denominator

Forms containing P(n)(e^x)/Q(m)(e^x)

Poisson's integral

Fresnel integrals

Chapter 8: Advanced Definite Integrals

Rational functions times trigonometric functions

Forms containing (x^2 - 2a sinx + a^2)-1

Forms containing (h sin a.x + x cos a.x)^-1

Forms containing Bessel functions

Chapter 9: Further Applications of the Theory of Residues

Counting zeros and poles of meromorphic functions

The argument principle

Rouche's Theorem

Simple-pole expansion of meromorphic functions

Infinite product expansion of entire functions

Chapter 10: Series Summation by Residues

Type of series considered

Summation of S1

Summation of S2

Summation of S3 and S4

Series with neither even nor odd terms

Series involving real zeros of entire functions

Series involving complex zeros of entire functions