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

Subject Matter

 * Complex Analysis

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