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

Subject Matter

 * Complex Analysis

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