Outline

# Mapping Diagrams to Visualize Complex Analysis

Mapping Diagrams to Visualize Complex Analysis illustrates the use of mapping diagrams in the analysis of functions of a complex variable.
This work was motivated initially by https://sites.google.com/site/visualcomplexanalysis/
Workshop material prepared for IV Nordic GeoGebra Conference, 20.-22. September, Copenhagen, Denmark
By Rokas Tamošiūnas (rokas.tamosiunas@ksu.lt)
I have organized an MAA mini-course on this topic: Joint Mathematics Meetings Denver, CO Jan. 15&17, 2020.
MAA Minicourse 2020: Visual Complex Analysis- GeoGebra Tools and Mapping Diagrams GeoGebra Book. https://www.geogebra.org/m/rsqxtq9t
A guide on how to explore complex integration in other ways with GeoGebra can be found at: http://goo.gl/1aToaT
Another publication with a little bit different approach can be found at: http://ggijro1.files.wordpress.com/2013/10/visualizing_complex_integral_final1.pdf .

## Table of Contents

### Introduction to Complex Numbers and Arithmetic

### Linear and Quadratic Functions and Equations

- Table and Mapping Diagram for A Complex Linear Function
- Complex Linear Mapping Diagram II: f(z) = Az
- Complex Linear Mapping Diagram III (lines)
- Complex Linear Mapping Diagrams I (Circles)
- Complex Mapping Diagrams for Quadratic Function and Solving Real Quadratic Equations
- Mapping Diagrams for Complex Quadratic Functions & Equations
- Mapping Diagram Solving A Quadratic Equation(Real and Complex)
- Complex Variable Function Mapping Diagrams and Solving Real Cubic Equations

### Mapping Diagrams to Visualize Elementary Functions

- Mapping Diagrams: Complex Analysis -functions with complex parameters. circles
- Mapping Diagrams: Complex Analysis-functions with lines or circles
- Mapping Diagram: Complex Analysis. Realign Target.
- Complex Linear Mapping Diagram f(z) = Bz; parametric curve
- Moebius Transformation: MD circle with spherical projection
- Complex Mapping Diagram for Moebius Function (circles)
- Mapping diagram for Moebius Function
- Mapping diagram for Moebius Transformation: roots and poles, lines and circles
- Mapping Diagram for Complex Power Function (Circles)
- Mapping Diagram for z^n with Curve on Riemann Surface
- Mapping Diagram for z^n
- Complex Mapping diagram with homotopy
- Mapping Diagram Powers Homotopy
- Mapping Diagrams for Quartic Equations
- Mapping Diagrams for Visualizing Roots for Polynomials
- Complex plane transformations: mapping a line
- Complex square mapping between two planes

### Limits and Continuity for Complex Functions

### Mapping Diagrams to Visualize Differentiation and Applications

### Mapping Diagrams to Visualize Integration

- Complex Line Integral: Definition and F.T. Examples
- Mapping Diagram for Cauchy Integral Formula
- Complex Integration from list
- Integral Examples from List
- Mapping Diagram: Complex Integration Parametric Function
- Mapping Diagram for Complex Integration over A Circle
- Mapping Diagram: Complex Integration of 1/z
- Complex Integration: z^2 over line segment
- Complex Integration: z^2 over curve
- The Winding Number of a Curve about a Point

### Series: Taylor and Laurent