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

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