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

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

Published $\text {1983}$, Cambridge University Press

ISBN 0-521-28763-4.

### 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
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

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