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Solutions to Exercises of the bookÂ
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
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