Reed College

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September 19, 2016

**Complex Variables: Mapping Diagrams for **

Visualizing Complex Arithmetic and Functions

Dynamically with GeoGebra

Martin Flashman

Professor of Mathematics

Humboldt State University

http://flashman.neocities.org/Presentations/MD.Reed.CV.9_19_16.html

¤

September 19, 2016

Visualizing Complex Arithmetic and Functions

Dynamically with GeoGebra

Part I Mapping Diagrams for Real Functions Complex Arithmetic |
Part IIComplex Functions |

Martin Flashman

Professor of Mathematics

Humboldt State University

http://flashman.neocities.org/Presentations/MD.Reed.CV.9_19_16.html

Here are some examples of

Example 3.1.1.1Mapping Diagram for Complex Linear Function from a Table (using points on a lattice for sampled domain.)

This example shows a linear function $f(z) = a + bz$ where $a$ and $b \in \mathbb{C}$

Move $a$ and $b$ in the plane to change the value of these complex number parameters.Move the "lattice" point in the plane to change the position of the lattice being used for the data in the table and on the mapping diagram.

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**$s_a(z) = a + z$**.**

3.1.2

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.**

http://flashman.neocities.org/Presentations/MD.Reed.CV.9_19_16.html

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