1. |
The complex plane | |||||||
  | 1.1 | The complex numbers and the complex plane | ||||||
  |   | 1.1.1 | A formal view of the complex numbers | |||||
  | 1.2 | Some geometry | ||||||
  | 1.3 | Subsets of the plane | ||||||
  | 1.4 | Functions and limits | ||||||
  | 1.5 | The exponential, logarithm, and trigonometric functions | ||||||
  | 1.6 | Line integrals and Green's theorem | ||||||
2. | Basic properties of analytic functions | |||||||
  | 2.1 | Analytic and harmonic functions; the Cauchy-Riemann equations | ||||||
  |   | 2.1.1 | Flows, fields, and analytic functions | |||||
  | 2.2 | Power series | ||||||
  | 2.3 | Cauchy's theorem and Cauchy's formula | ||||||
  |   | 2.3.1 | The Cauchy-Goursat theorem | |||||
  | 2.4 | Consequences of Cauchy's formula | ||||||
  | 2.5 | Isolated singularities | ||||||
  | 2.6 | The residue theorem and its application to the evaluation of definite integrals | ||||||
3. | Analytic functions as mappings | |||||||
  | 3.1 | The zeros of an analytic function | ||||||
  |   | 3.1.1 | The stability of solutions of a system of linear differential equations | |||||
  | 3.2 | Maximum modulus and mean value | ||||||
  | 3.3 | Linear fractional transformations | ||||||
  | 3.4 | Conformal mapping | ||||||
  |   | 3.4.1 | Conformal mapping and flows | |||||
  | 3.5 | The Riemann mapping theorem and Schwarz-Christoffel transformations | ||||||
4. | Analytic and harmonic functions in applications | |||||||
  | 4.1 | Harmonic functions | ||||||
  | 4.2 | Harmonic functions as solutions to physical problems | ||||||
  | 4.3 | Integral representations of harmonic functions | ||||||
  | 4.4 | Boundary-value problems | ||||||
  | 4.5 | Impulse functions and the Green's function of a domain | ||||||
5. | Transform methods | |||||||
  | 5.1 | The Fourier transform: basic properties | ||||||
  | 5.2 | Formulas Relating u and û | ||||||
  | 5.3 | The Laplace transform | ||||||
  | 5.4 | Applications of the Laplace transform to differential equations | ||||||
  | 5.5 | The Z-Transform | ||||||
  |   | 5.5.1 | The stability of a discrete linear system | |||||
Appendix 1. | The stability of a discrete linear system | |||||||
Appendix 2. | A Table of Conformal Mappings | |||||||
Appendix 3. | A Table of Laplace Transforms | |||||||
  | Solutions to Odd-Numbered Exercises | |||||||
  | Index | |||||||