Book:H.A. Priestley/Introduction to Complex Analysis/Revised Edition

From ProofWiki
Jump to navigation Jump to search

H.A. Priestley: Introduction to Complex Analysis (Revised Edition)

Published $1990$, Oxford Science Publications

ISBN 0-19-853428-0.


Subject Matter


Contents

Preface to the revised edition (Oxford, November 1989)
Preface to the first edition (Oxford, March 1985)
Notation and terminology
1. The complex plane
Complex numbers
Open and closed sets in the complex plane
Limits and continuity
Exercises
2. Holomorphic functions and power series
Holomorphic functions
Complex power series
Elementary functions
Exercises
3. Prelude to Cauchy's theorem
Paths
Integration along paths
Connectedness and simple connectedness
Properties of paths and contours
Exercises
4. Cauchy's theorem
Cauchy's theorem, Level I
Cauchy's theorem, Level II
Logarithms, argument, and index
Cauchy's theorem revisited
Exercises
5. Consequences of Cauchy's theorem
Cauchy's formulae
Power series representation
Zeros of homomorphic functions
The Maximum-modulus theorem
Exercises
6. Singularities and multifunctions
Laurent's theorem
Singularities
Mesomorphic functions
Multifunctions
Exercises
7. Cauchy's residue theorem
Cauchy's residue theorem
Counting zeros and poles
Calculation of residues
Estimation of integrals
Exercises
8. Applications of contour integration
Improper and principal-value integrals
Integrals involving functions with a finite number of poles
Integrals involving functions with infinitely many poles
Deductions from known integrals
Integrals involving multifunctions
Evaluation of definite integrals: summary
Summation of series
Exercises
9. Fourier and Laplace transforms
The Laplace transform: basic properties and evaluation
The inversion of Laplace transforms
The Fourier transform
Applications to differential equations, etc.
Appendix: proofs of the Inversion and Convolution theorems
Convolutions
Exercises
10. Conformal mapping and harmonic functions
Circles and lines revisited
Conformal mapping
Möbius transformations
Other mappings: powers, exponentials, and the Joukowski transformation
Examples on building conformal mappings
Holomorphic mappings: some theory
Harmonic functions
Exercises
Supplementary exercises
Bibliography
Notation index
Subject index