28th International Conference on Technology in Collegiate Mathematics

2016

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March 12, 2016

**Mapping Diagrams for Complex Variable Functions **

Visualized Dynamically with GeoGebra

1:30 p.m. - 2:00 p.m.

Martin Flashman

Professor of Mathematics

Humboldt State University

http://users.humboldt.edu/flashman/Presentations/MD.ICTCM.CV.3_12_16.html

2016

¤

March 12, 2016

Visualized Dynamically with GeoGebra

1:30 p.m. - 2:00 p.m.

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

Martin Flashman

Professor of Mathematics

Humboldt State University

http://users.humboldt.edu/flashman/Presentations/MD.ICTCM.CV.3_12_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.**

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.

http://users.humboldt.edu/flashman/Presentations/MD.ICTCM.CV.3_12_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