Table of contents - Complex analysis

This is a table of contents for my notes on complex analysis. Topics are mainly those used in physics.

Complex numbers

• Complex arithmetic
• Sums, products and reciprocals of complex numbers
• Complex plane
• Reflection of complex number through a line
• Binomial theorem for complex numbers
• Complex matrix products
• Hermitian matrix theorems
• Complex numbers in polar form
• Argument of a complex number
• Scalar and vector products of complex numbers
• Complex exponentials and Euler's equation
• Powers of complex numbers and De Moivre's formula
• Sums of multiple angles
• Roots of complex numbers
• Sets in the complex plane - terminology
• Continuum sets
• Constant function on a domain
• Stereographic projection and the Riemann sphere
• Stereographic projection of circles

Analytic functions

• Functions of a complex variable
• Inversion mapping
• Exponential mapping
• Joukowski mapping
• Limits of complex functions
• Differentiable complex functions
• L'hopital's rule
• Cauchy-Riemann equations
• Analytic and entire functions
• Constant complex functions
• Harmonic functions
• Harmonic conjugates

Elementary functions

• Factoring polynomials
• Taylor form of polynomials
• Partial fractions
• Partial fractions - examples
• Poles of rational functions
• Equality of two rational functions
• Residues
• Complex exponential, trig and hyperbolic functions
• Reciprocal exponents
• Linearly independent exponentials
• Alternative proof of a trig identity
• Complex logarithms
• Branches of logarithms
• Boundary value problems - circles
• Boundary value problems - wedges
• Complex powers
• Analytic branches of roots
• Inverse trigonometric functions
• Inverse secant

Complex integration

• Smooth curves
• Contours
• Contour integrals
• Estimating contour integrals
• Continuously deformable loops
• Cauchy's integral theorem
• Cauchy's integral formula
• Cauchy estimates
• Cauchy estimates - examples
• Liouville's theorem
• Maximum modulus principle
• Poisson integral formula
• Poisson integral formula for the half-plane
• Harnack's inequality

Series representations for analytic functions

• Convergent series
• Ratio test for series
• Jumping frog problem
• Pointwise and uniform convergence
• Taylor series for complex functions
• Generalized binomial theorem
• Convergence of Taylor series
• Recentering polynomials
• Cauchy product of Taylor series
• Remainder in Maclaurin series
• Error estimates in Taylor series
• Circle of convergence for power series

Complex functions

• Analytic continuation
• Branch points and cuts in the complex plane

Complex integration

• Contour integration
• Contour integration around a branch cut