# Book:E.G. Phillips/Functions of a Complex Variable/Eighth Edition

## E.G. Phillips: Functions of a Complex Variable (8th Edition)

Published $\text {1957}$, Oliver & Boyd Ltd.

### Contents

PREFACE TO THE EIGHTH EDITION
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 $\text {I}$
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 $\text {II}$
SOME SPECIAL TRANSFORMATIONS
The Transformations $w = z^n$
$w = z^2$
$w = \sqrt z$
$w = \map {\tan^2} {\tfrac 1 4 \pi \sqrt z}$
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 $\text {III}$
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 $\text {IV}$
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 $\text {V}$
MISCELLANEOUS EXAMPLES
INDEX

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