Complex analysis

.

Table of contents

1. Introduction

   • Complex numbers

   • Complex algebra

2. Complex plane

   • Trigonometric form

   • Polar form

3. Complex functions

   • Colour wheel graph

   • Polynomials

   • Rational functions

   • Exponential function

   • Logarithm

   • Trigonometric and hyperbolic functions

4. Holomorphic functions

   • Complex differentiation

   • Cauchy-Riemann conditions

5. Complex integration

   • Contours

   • Contour integral

   • Cauchy’s theorem

6. Fractals

   • Koch curve and snowflake

   • Mandelbrot set

Further reading and references

• Prof Velisa Vesovic’s lecture notes on Maths Methods 2 (available internally on ESESIS)

• Elias Wegert’s book Visual Complex Functions

• Dr Sheehan Olver’s lecture notes on Applied Complex Analysis and his Julia script ComplexPhasePortrait.jl motivated many visualisations and our Python implementation of them

• Ablowitz, M.J., Fokas, A.S. and Fokas, A.S., 2003. Complex variables: introduction and applications. Cambridge University Press.

• Arfken, G.B. and Weber, H.J., 1999. Mathematical methods for physicists.

• Ruban, A.I. and Gajjar, J.S., 2014. Fluid Dynamics: Classical Fluid Dynamics. Oxford University Press, USA.