"Revitalizing Complex Analysis" ¤

January 9, 2016

Linear Fractional Transformations.

Martin Flashman

Professor of Mathematics

Humboldt State University

Link for these notes:

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

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-

Mapping Diagrams and Graphs of Real Linear Functions

Here are some examples of

Using the "r" slider to change the radius of the circle in the domain for the mapping diagram.

A complex

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

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.

Thanks for participating

AMATYC Webinar M Flashman Using Mapping Diagrams to Understand Trig Functions (YouTube)

Martin Flashman ...Solving Linear Equations Visualized with Mapping Diagrams (YouTube)

Martin Flashman ...Partial Derivatives: An Introduction Using Mapping Diagrams (You Tube)

Martin Flashman ...Solving Linear Equations Visualized with Mapping Diagrams (YouTube)

Martin Flashman ...Partial Derivatives: An Introduction Using Mapping Diagrams (You Tube)

A Reference and Resource Book on Function Visualizations Using Mapping Diagrams