Mapping Diagrams to Visualize Complex Analysis

Martin Flashman, Dec 9, 2015

Mapping Diagrams to Visualize Complex Analysis illustrates the use of mapping diagrams in the analysis of functions of a complex variable.

Table of Contents

  1. Introduction and Linear Functions
    1. Complex Plane
    2. Complex Arithmetic
    3. Complex Addition
    4. Complex Multiplication
    5. Complex 1/z
    6. Table and Mapping Diagram for A Complex Linear Function
    7. Complex Linear Mapping Diagram f(z) = az
    8. Mapping Diagram for Complex Linear Function (Circles)
    9. Complex Mapping Diagram of Linear Function (lines)
  2. Mapping Diagrams to Visualize Elementary Functions
    1. Complex Linear Mapping Diagram f(z) = bz; parametric curve
    2. Complex MD Moebius function circle
    3. Complex Mapping Diagram for Moebius Function (circles)
    4. Mapping diagram for Moebius Function
    5. Mapping Diagram for Complex Power Function (Circles)
    6. Mapping Diagram for z^n with Curve on Riemann Surface
    7. Mapping Diagram for z^n
  3. Limits for Complex Functions
    1. Limits for Complex Functions
    2. Mapping Diagram:Limits for Complex Functions
    3. Limits for Complex Functions
  4. Mapping Diagrams to Visualize Differentiation
    1. Mapping Diagram Visualizing Complex Differential
    2. Mapping Diagram Visualizing Complex Derivative:Powers
  5. Mapping Diagrams to Visualize Integration
    1. Mapping Diagram for Complex Integration of 1/z

1.

Introduction and Linear Functions

  • 1. Complex Plane
  • 2. Complex Arithmetic
  • 3. Complex Addition
  • 4. Complex Multiplication
  • 5. Complex 1/z
  • 6. Table and Mapping Diagram for A Complex Linear Function
  • 7. Complex Linear Mapping Diagram f(z) = az
  • 8. Mapping Diagram for Complex Linear Function (Circles)
  • 9. Complex Mapping Diagram of Linear Function (lines)

1.1.

Complex Plane
– Martin Flashman

1.2.

–

1.3.

–

1.4.

–

1.5.

–

1.6.

–

1.7.

–

1.8.

–

1.9.

–

2.

Mapping Diagrams to Visualize Elementary Functions

  • 1. Complex Linear Mapping Diagram f(z) = bz; parametric curve
  • 2. Complex MD Moebius function circle
  • 3. Complex Mapping Diagram for Moebius Function (circles)
  • 4. Mapping diagram for Moebius Function
  • 5. Mapping Diagram for Complex Power Function (Circles)
  • 6. Mapping Diagram for z^n with Curve on Riemann Surface
  • 7. Mapping Diagram for z^n

2.1.

Complex Linear Mapping Diagram f(z) = bz; parametric curve

Complex Linear Mapping Diagram f(z) = bz on parametric curve.
Move b and t# to change the parameter b and t#.[br]Click on the boxes to show mapping diagram arrows .[br]Change r on the slider to change the domain parabola"z = r(t+t^2i)".
– Martin Flashman

2.2.

–

2.3.

–

2.4.

–

2.5.

–

2.6.

–

2.7.

–

3.

Limits for Complex Functions

  • 1. Limits for Complex Functions
  • 2. Mapping Diagram:Limits for Complex Functions
  • 3. Limits for Complex Functions

3.1.

Limits for Complex Functions

Limits for [math]f(z) = z^2+2z[/math] at z = i. SOS Prexercise: 2.89
– Martin Flashman

3.2.

–

3.3.

–

4.

Mapping Diagrams to Visualize Differentiation

  • 1. Mapping Diagram Visualizing Complex Differential
  • 2. Mapping Diagram Visualizing Complex Derivative:Powers

4.1.

Mapping Diagram Visualizing Complex Differential

Mapping Diagram Visualizing Complex Differential
– Martin Flashman

4.2.

–

5.

Mapping Diagrams to Visualize Integration

  • 1. Mapping Diagram for Complex Integration of 1/z

5.1.

Mapping Diagram for Complex Integration of 1/z

Mapping Diagram for Complex Integration of 1/z
– Martin Flashman

Information