Table of Contents
( \newcommand{\kernel}{\mathrm{null}\,}\)
Table of Contents
-
Licensing
-
Preface
-
1: Complex Algebra and the Complex Plane
- 1.1: Motivation
- 1.2: Fundamental Theorem of Algebra
- 1.3: Terminology and Basic Arithmetic
- 1.4: The Complex Plane
- 1.5: Polar Coordinates
- 1.6: Euler's Formula
- 1.7: The Exponential Function
- 1.8: Complex Functions as Mappings
- 1.9: The function arg(z)
- 1.10: Concise summary of branches and branch cuts
- 1.11: The Function log(z)
- 1.12: Inverse Euler formula
- 1.13: de Moivre's formula
- 1.14: Representing Complex Multiplication as Matrix Multiplication
-
2: Analytic Functions
- 2.1: The Derivative - Preliminaries
- 2.2: Open Disks, Open Deleted Disks, and Open Regions
- 2.3: Limits and Continuous Functions
- 2.4: The Point at Infinity
- 2.5: Derivatives
- 2.6: Cauchy-Riemann Equations
- 2.7: Cauchy-Riemann all the way down
- 2.8: Gallery of Functions
- 2.9: Branch Cuts and Function Composition
- 2.10: Appendix - Limits
-
3: Multivariable Calculus (Review)
-
4: Line Integrals and Cauchy’s Theorem
-
5: Cauchy Integral Formula
-
6: Harmonic Functions
-
7: Two Dimensional Hydrodynamics and Complex Potentials
-
8: Taylor and Laurent Series
-
9: Residue Theorem
-
10: Definite Integrals Using the Residue Theorem
-
11: Conformal Transformations
- 11.1: Geometric Definition of Conformal Mappings
- 11.2: Tangent vectors as complex numbers
- 11.3: Analytic functions are Conformal
- 11.4: Digression to harmonic functions
- 11.5: Riemann Mapping Theorem
- 11.6: Examples of conformal maps and excercises
- 11.7: Fractional Linear Transformations
- 11.8: Reflection and symmetry
- 11.9: Flows around cylinders
- 11.10: Solving the Dirichlet problem for harmonic functions
-
12: Argument Principle
-
13: Laplace Transform
- 13.1: A brief introduction to linear time invariant systems
- 13.2: Laplace transform
- 13.3: Exponential Type
- 13.4: Properties of Laplace transform
- 13.5: Differential equations
- 13.6: Table of Laplace transforms
- 13.7: System Functions and the Laplace Transform
- 13.8: Laplace inverse
- 13.9: Delay and Feedback
-
14: Analytic Continuation and the Gamma Function
-
Index
-
Detailed Licensing