Visualising

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)

Analytic Landscapes

\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

Mappings

Mappings

Domain colouring

Phase portraits

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) 

Implementation in the computer

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

Implementation in the computer

Basic Examples

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

Enhanced phase portraits: Level curves

f(z)=z

Phase

Modulus

Enhanced phase portraits: Level curves

f(z)=z

Phase

Modulus

Combined

Enhanced phase portraits: Modulus

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

Enhanced phase portraits: \(f(z)=\dfrac{z-1}{z^2+z+1}\)

More examples...

Roots of unity: \(z^n-1,\;n=2,3,\ldots,10\)

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.
a_n=\frac{1}{2\pi i}\displaystyle \oint_C \frac{f(z)dz}{(z-z_0)^{n+1}}\quad (n=0,1,2,\ldots)
b_n=\frac{1}{2\pi i}\displaystyle \oint_C \frac{f(z)dz}{(z-z_0)^{-n+1}}\quad (n=1,2,\ldots)
\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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