# Open-source online tools to visualise and explore complex functions

### with domain colouring ðŸŒˆ

Juan Carlos Ponce Campuzano

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1. What is domain colouring?
2. How can it be implemented in the computer?
3. How can we use it to explore complex functions?
4. Open-source online tools

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Complex functions

f:\mathbb C \rightarrow \mathbb C

live in a 4-dimensional space

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### Methods to visualize complex functions

- Domain colouring

- Real and Imaginary components

- Analytic Landscapes

- Mappings

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## 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 colour determined by the value $$f(z)$$.

### The colour wheel

• H = Phase
• S = 1
• B = 1

Hue , Saturation & Brightness

(HSB)Â

# Phase portraits

## 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

## Online tools

Thank you!

Online resources:

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

Contact:

j.ponce@uq.edu.au

Slides: reveal.js

#### Open-source online tools to visualise and explore complex functions with domain colouring

By Juan Carlos Ponce Campuzano

# Open-source online tools to visualise and explore complex functions with domain colouring

Presentation for the Delta Conference 2021. https://www.herengadelta.org/

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