ProofWiki:Books/E.G. Phillips/Functions of a Complex Variable

E.G. Phillips: Functions of a Complex Variable

Published 1940, 8th Edition 1957, Oliver & Boyd Ltd..

Subject Matter

• Complex Analysis

Contents

PREFACE

FUNCTIONS OF A COMPLEX VARIABLE

Complex Numbers

Sets of Points in the Argand Diagram

Functions of a Complex Variable

Regular Functions

Conjugate Functions

Power Series

The Elementary Functions

Many-valued Functions

Examples I

CONFORMAL REPRESENTATION

Isogonal and Conformal Transformations

Harmonic Functions

The Bilinear Transformation

Geometrical Inversion

The Critical Points

Coaxal Circles

Invariance of the Cross-Ratio

Some special Möbius' Transformations

Examples II

SOME SPECIAL TRANSFORMATIONS

The Transformations $w = z^n$

$w = z^2$

$w = \sqrt z$

$w = \tan^2 \left({\tfrac 1 4 \pi \sqrt z}\right)$

Combinations of $w = z^a$ with Möbius' Transformations

Exponential and Logarithmic Transformations

Transformations involving Confocal Conics

$z = c \sin w$

Joukowski's Aerofoil

Tables of Important Transformations

Schwarz-Christoffel Transformation

Examples III

THE COMPLEX INTEGRAL CALCULUS

Complex Integration

Cauchy's Theorem

The Derivatives of a Regular Function

Taylors Theorem

Liouville's Theorem

Laurent's Theorem

Zeros and Singularities

Rational Functions

Analytic Continuation

Poles and Zeros of Meromorphic Functions

Rouché's Theorem

The Maximum-Modulus Principle

Examples IV

THE CALCULUS OF RESIDUES

The Residue Theorem

Integration round the Unit Circle

Evaluation of Infinite Integrals

Jordan's Lemma

Integrals involving Many-valued Functions

Integrals deduced from Known Integrals

Expansion of a Meromorphic Function

Summation of Series

Examples V

MISCELLANEOUS EXAMPLES

INDEX