Visualizing Complex Variable Functions with Mapping Diagrams: Linear Fractional Transformations.

"Revitalizing Complex Analysis" ¤
January 9, 2016
Visualizing Complex Variable Functions with Mapping Diagrams:
Linear Fractional Transformations.

Martin Flashman
Professor of Mathematics
Humboldt State University

Link for these notes:
http://flashman.neocities.org/Presentations/MD.JMM.1_9_16.html

Abstract: Crucial to understanding much about complex variable functions is a sound comprehension of core linear fractional transformations. In a recent undergraduate complex variables course the author used GeoGebra 5.0 to create mapping diagrams for these functions in a three dimensional setting. The diagrams are modeled after mapping diagrams for real variable functions. Several visual features of these functions and their complex integrals will be illustrated that can add to the understanding of these core functions. The diagrams will be available for use over the internet.
Background and References to other work on Mapping Diagrams for real variables ¤

1.1.Background: Mapping Diagrams for Real Functions . ¤

What is a mapping diagram?
Introduction and simple examples from the past: Napiers Logarithm

Real Linear Functions.
Understanding functions using tables. mapping diagrams and graphs.
Functions: Tables, Mapping Diagrams, and Graphs

1.2. Real Linear Functions. ¤
Real Linear functions are the key to understanding calculus.
Linear functions are traditionally expressed by an equation like : $f(x)= mx + b$.
Mapping diagrams for real linear functions have one simple unifying feature- the focus point, determined by the numbers $m$ and $b$, denoted here by $[m,b]$.
Mapping Diagrams and Graphs of Real Linear Functions

Visualizing real linear functions using mapping diagrams and graphs.
This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com
Notice how points on the graph pair with arrows and points on the mapping diagram.

2.1. Complex Functions ¤

Functions of complex variables can be visualized in a 3 dimensional figure by mapping diagrams between parallel complex planes.
Here are some examples of linear functions, linear fractional (Moebius), and power functions I have built using GeoGebra .

2.1.1 A complex linear function can be visualized as a mapping from C to C.

Mapping Diagram for Complex Linear Function (using points on circles for sampled domain.)
This example shows a linear function on a single complex number and points on a circle in the domain complex plane.
Using the "r" slider to change the radius of the circle in the domain for the mapping diagram.

2.1.2 Complex Mapping Diagram of Linear Function (lines) ¤
A complex linear function can be visualized as a mapping from C to C.
This example shows a linear function on a single complex number and points on a line in the complex plane.

You can move the point z_#, change the slider to move the line, and enter new function.

2.1.3 Visualizing Complex Linear Fractional Transformations [Moebius Functions] with Mapping Diagrams ¤

You can move a,b,and c in the Graphics window to change the complex parameters for the Moebius function.

The sliders adjust the height of the second complex plane and the radius of the circle for the mapping diagram arrow source.

2.1.4 Bonus! Visualizing Complex Power Functions with Mapping Diagrams ¤

The sliders adjust the height of the second complex plane, the power, n, and the radius of the circle for the mapping diagram arrow source.

Thanks for participating¤.

Questions?

The End!

http://flashman.neocities.org/Presentations/MD.JMM.1_9_16.html

References:¤

AMATYC Webinar Martin Flashman - Using Mapping Diagrams to Understand Functions (YouTube)

Mapping Diagrams from A(lgebra) B(asics) to C(alculus) and D(ifferential) E(quation)s.
A Reference and Resource Book on Function Visualizations Using Mapping Diagrams

The Sensible Calculus Program