# Complex Functions

Enhanced phase portraits

1. What domain colouring is
2. How it can be implemented in the computer
3. How we can use it to explore complex functions

Complex functions

f:\mathbb C \rightarrow \mathbb C

live in a 4-dimensional space

## Methods to visualize complex functions

• Real and Imaginary components
• Analytic Landscapes
• Mappings
• Domain colouring

## Real and Imaginary components

f(z) = \text{Re}(z) + i \, \text{Im}(z)
f(z) = z^2
f(z) = x^2-y^2 + 2xy\,i
f(z) = x^2-y^2 + 2xy\,i
\text{Re}(z^2)
\text{Im}(z^2)

\big| f(z) \big|

### A historical analytic landscape of $$\big|\Gamma(z)\big|$$ from 1909

Funktionentafeln mit Formeln und Kurven by Eugene Jahnke & Fritz Emde

# Domain colouring

## Domain colouring

1. Assign a colour to every point in the complex plane.
2. Colour the domain of $$f$$ by painting the location $$z$$ with the color determined by the value $$f(z)$$.

### The colour wheel

• H = Phase
• S = 1

• B = 1

Hue , Saturation & Brightness

(HSB)

• Mathematica
• MATLAB
• Python
• Java
• C++
• GeoGebra
• JavaScript
• CindyJS
• p5.js

# Basic Examples

## Phase portrait

f(z)=z\\ [-2,2] \times [-2,2]

## Phase portrait

f(z)=\dfrac{1}{z}\\ [-2,2] \times [-2,2]

## Phase portrait

f(z)=\dfrac{z-1}{z^2+z+1}\\ [-2,2] \times [-2,2]

## Enhanced phase portraits

• H = Phase
• S = 1
• B = $$\log\big|f\big|- \lfloor \log |f| \rfloor$$

Elias Wegert's work from 2012

=\text{Phase}-\lfloor \text{Phase} \rfloor

f(z)=z

Phase

Modulus

f(z)=z

Phase

Modulus

Combined

f(z)=z
f(z)=1/z

# More examples...

## Multiplicity of zeros & Order of poles

f(z)=(z+1)z^2(z-1)^3
g(z)=1/f(z)

## Analytic vs Non-Analytic functions

f(z)=\frac{3}{2}z(1+iz)
g(z)=\frac{3}{2}z(1-i\overline{z})

## Laurent series

f(z)=\displaystyle\sum_{n=0}^{\infty} a_n(z-z_0)^n+\displaystyle\sum_{n=1}^{\infty}\frac{b_n}{(z-z_0)^n},
R_1 <|z-z_0|< R_2.
\dfrac{1-\cosh(z)}{z^3}

### Poles of order $$m$$:  $$\exists m \geq 1, \, b_m\neq 0$$ and $$b_k=0$$ for $$k>m$$.

\dfrac{\exp(2z)}{(z-1)^2}
\dfrac{\sinh(z)}{z^4}

## Removable singularities: If $$b_n=0,\; \forall n$$

\dfrac{\sin(z)}{z}
\dfrac{z}{e^z-1}

## Essential singularities: If $$b_k\neq0$$ for infinitely many $$k$$

\sin \left(\dfrac{1}{z}\right)
\exp \left(\dfrac{1}{z}\right)

## Zooming in

$$f(z)=\exp\left(\dfrac{1}{z}\right)$$

# Other colour schemes

### $$f(z)=0.926(z+0.073857 z^5+0.0045458 z^9)$$

Discrete HSV

RGB

B&W

Thank you!

Online resources:

https://www.dynamicmath.xyz/domain-coloring/

https://complex-analysis.com/

Contact:

j.ponce@uq.edu.au

Slides: reveal.js

#### Visualising Complex Functions

By Juan Carlos Ponce Campuzano

# Visualising Complex Functions

A brief introduction to domain coloring to visualize and study complex functions.

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