# Book:Ian Stewart/Complex Analysis (The Hitchhiker's Guide to the Plane)

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## Ian Stewart and David Tall:

## Ian Stewart and David Tall: *Complex Analysis (The Hitchhiker's Guide to the Plane)*

Published $1983$, **Cambridge University Press**

- ISBN 0-521-28763-4.

### Subject Matter

### Contents

*Preface**Acknowledgement*

**0 The origins of complex analysis, and a modern viewpoint**- 1. The origins of complex numbers
- 2. The origins of complex analysis
- 3. The puzzle
- 4. A modern view

**1 Algebra of the Complex Plane**- 1. Construction of the complex numbers
- 2. The $x + iy$ notation
- 3. A geometric interpretation
- 4. Real and imaginary parts
- 5. The modulus
- 6. The complex conjugate
- 7. Polar coordinates
- 8. The complex numbers cannot be ordered
- Exercises 1

**2 Topology of the complex plane**- 1. Open and closed sets
- 2. Limits of functions
- 3. Continuity
- 4. Paths
- 5. The Paving Lemma
- 6. Connectedness
- Exercises 2

**3 Power Series**- 1. Sequences
- 2. Series
- 3. Power series
- 4. Manipulating power series
- 5. Appendix
- Exercises 3

**4 Differentiation**- 1. Basic results
- 2. The Cauchy-Riemann equations
- 3. Connected sets and differentiability
- 4. Hybrid functions
- 5. Power series
- 6. A glimpse into the future
- Exercises 4

**5. The exponential function**- 1. The exponential function
- 2. Real exponentials and logarithms
- 3. Trigonometric functions
- 4. The analytic definition of $\pi$
- 5. The behaviour of real trigonometric functions
- 6. Complex exponential and trigonometric functions are periodic
- 7. Other trigonometric functions
- 8. Hyperbolic functions
- Exercises 5

**6. Integration**- 1. The real case
- 2. Complex integration along smooth paths
- 3. The length of a smooth path
- 4. Contour integration
- 5. The Fundamental Theorem of Contour Integration
- 6. The Estimation Lemma
- 7. Consequences of the Fundamental Theorem
- Exercises 6

**7. Angles, logarithms, and the winding number**- 1. Radian measures of angles
- 2. The argument of a complex number
- 3. The complex logarithm
- 4. The winding number
- 5. The winding number as an integral
- 6. The winding number round an arbitrary point
- 7. Components of the complement of a path
- 8. Computing the winding number by eye
- Exercises 7

**8 Cauchy's Theorem**- 1. The Cauchy Theorem for a triangle
- 2. Existence of an antiderivative in a star-domain
- 3. An example - the logarithm
- 4. Local existence of an antiderivative
- 5. Cauchy's Theorem
- 6. Applications of Cauchy's Theorem
- 7. Simply connected domains
- Exercises 8

**9 Homotopy versions of Cauchy's Theorem**- 1. Integration along arbitrary paths
- 2. The Cauchy Theorem for a boundary
- 3. Homotopy
- 4. Fixed end point homotopy
- 5. Closed path homotopy
- 6. The Cauchy Theorems compared
- Exercises 9

**10 Taylor series**- 1. Cauchy's integral formula
- 2. Taylor series
- 3. Morera's Theorem
- 4. Cauchy's Estimate
- 5. Zeros
- 6. Extension functions
- 7. Local maxima and minima
- 8. The Maximum Modulus Theorem
- Exercises 10

**11 Laurent series**- 1. Series involving negative powers
- 2. Isolated singularities
- 3. Behaviour near an isolated singularity
- 4. The extended complex plane, or Riemann sphere
- 5. Behaviour of a differentiable function at $\infty$
- 6. Meromorphic functions
- Exercises 11

**12 Residues**- 1. Cauchy's residue theorem
- 2. Calculating residues
- 3. Evaluation of definite integrals
- 4. Summation of series
- 5. Counting zeroes
- Exercises 12

**13 Conformal transformations**- 1. Real numbers modulo $2 \pi$
- 2. Conformal transformations
- 3. Möbius mappings
- 4. Potential theory
- Exercises 13

**14 Analytic continuation**- 1. The limitations of power series
- 2. Comparing power series
- 3. Analytic continuation
- 4. Multiform functions
- 5. Riemann surfaces
- 6. Complex powers
- 7. Conformal mapping using multiform functions
- 8. Contour integration of multiform functions
- 9. The road goes ever on ...
- Exercises 14

- Index