Contents

• Contents
• Complex number arithmetic
  • complex numbers in the real plane via the Argand diagram
  • polar and cartesian coordinates
  • absolute value, phase, modulus
  • stereographic projection
  • Smith Chart
  • graphical representation of complex functions
• Functions of a complex variable
  • one-to-one and invertible
  • Möbius transformations represented as cross ratios
  • Möbius transformations representable as 2x2 matrices
  • eigenvalues and eigenvectors of a Möbius transformation
  • hyperbolic, parabolic, elliptic transformations
• The derivative of a function of a complex variable
  • Cauchy-Riemann equations
  • harmonic functions of real variables
  • the Schwartz derivative
• Iterated functions
  • fixed points and their stability
  • Mandelbrot set for second degree polynomials
• Contour Integrals
  • evaluation of integrals by using the residues at poles
  • existence of derivatives of all orders
  • Liouville's theorem: a bounded analytic function is constant
  • The maximum modulus principle
  • Schwartz's lemma
  • residues and the stability of fixed points
  • representation of a function by a power series
  • the monodromy principle
• Periodic functions
  • Eisenstein series
  • Weierstrass $\wp$ function
  • restraints on periodic functions
  • the differential equation for $\wp$
  • the Jacobi functions sn, cn, and dn
• Mapping theorems
  • interpolation
  • mapping the real line to a polygon
  • functions with boundary values
• Complex functions solving differential equations
  • kinds of equations
  • the differential calculus of matrices
  • linear matrix differential equations
  • the matrizant
  • uniqueness and periodicity
  • the coefficient matrix as a tensor sum
• Second order differential equations
  • polar coordinates and Prüfer's transformation
  • projective coordinates and Ricatti's equation
  • solving a Schwartz derivative
• Functions of mathematical physics
  • survey of equations
  • Bessel functions
  • Legendre polynomials
  • Mathieu functions
  • Airy functions for a linear potential
  • Dirac harmonic oscillator
• Sturm-Liouville boundary conditions
  • self adjoint equations and the symplectic metric
• Bibliography