"Revitalizing Complex Analysis" ¤

January 9, 2016

Linear Fractional Transformations.

Martin Flashman

Professor of Mathematics

Humboldt State University

http://users.humboldt.edu/flashman/Presentations/JMM2016/MD.JMM.CV.1_9_16.3.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 . ¤

1.1.1 What is a mapping diagram?

Introduction and simple examples from the past: Napiers Logarithm

See:
http://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms

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**Logarithms: The Early History of a Familiar Function - John Napier Introduces Logarithms* by Kathleen M. Clark (The Florida State University) and Clemency Montelle (University of Canterbury)

1.1.2 Real Functions. ¤

Understanding Real 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

Notes:

**When $a=0$ and**$b \in \mathbb{R}$**then $f$ is a dilation and the arrows lie on a section of a cone. The vertex of the cone is the focus point with axis orthogonal to the complex planes at $0$.**

**When $a=0$ and $b = e^{i\Theta} $ then $f$ is a rotation isometry about 0 by $\Theta$ radians and the arrows lie on a section of an hyperboloid of one sheet between circles of equal radii centered at $0$.****When $b = 1$ then $f$ is a translation isometry by $a$ and the arrows lie on a cylinder between circles of equal radii, one on the source centered at the $0$ and the other on the target centered at $a$.**

**In general one can understand $f$ as a composition of these three types of functions,**$f = s_a \circ r_{Arg(b)} \circ m_{|b|}$**where**$m_{|b|}(z) = |b|z$**,**$r_{Arg(b)}(z) = e^{Arg(b) i} z$**, and**$s_a(z) = a + z$**.**

2.1.2

In the following figure for $f(z) = bz$ the mapping diagram is

**When $b=c=0$ and**$b=1$**then $f$ is complex inversion and the circular based arrows cross on a line above the real axis.**

**When $a=1$ and $c=0 $ then $f$ is the composition of complex inversion followed by translation by the complex number $b$.****When $a=1$ and $b=0$ then $f$ is a translation isometry by $c$, followed by complex inversion.**

The following mapping diagram for a general Moebius function demonstrates the projective connection of the image complex plane to the Riemann sphere.

To change the complex parameters for the Moebius function y

Notice that if the circle passes through a pole of the function, the image is a line corresponding to a circle on the Riemann Sphere passing through the point at infinity.

Thanks for attending

M. Flashman GeoGebra Book [in development]: Mapping Diagrams to Visualize Complex Analysis http://ggbtu.be/bNi69jyKs

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

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