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

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## H.A. Priestley:

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

Published $\text {1990}$, **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
- MÃ¶bius 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**

## Further Editions

## Source work progress

- 1990: H.A. Priestley:
*Introduction to Complex Analysis*(Revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.5$ Subsets of the complex plane