Complex Analysis: A visual and interactive introduction

Juan Carlos Ponce Campuzano, Jan 31, 2019

Here you can find the GeoGebra applets from the online book [url]https://complex-analysis.github.io/[/url] If you want to support this project, you can do it using this link: [url]https://www.patreon.com/jcponce[/url] ∞ Thanks!

Table of Contents

  1. Preliminaries
    1. Complex numbers: cartesian representation
    2. Complex numbers: Polar and exponential forms
    3. Complex arithmetic: geometric interpretation
    4. Complex Roots
    5. Argument and Principal argument
    6. Multiplication of complex numbers as stretching (squeezing) and rotation
    7. Polar multiplication of complex numbers
    8. Addition-Subtraction of complex numbers
    9. Explore quadratic equations
    10. Explore cubic equation
  2. Functions
    1. Real and Imaginary components
    2. Riemann sphere
    3. Complex differentiation
  3. Mappings
    1. Mappings
    2. Inversion 1/z
    3. The mapping 1/z
    4. The Joukowsky map
    5. Complex mappings
  4. Riemann surfaces: Examples
    1. Real component: z^(1/2)
    2. Imaginary component: z^(1/2)
    3. Real component: z^(1/3)
    4. Imaginary component: z^(1/3)
    5. Real component: (1-z^2)^(1/2)
    6. Imaginary component: (1-z^2)^(1/2)
    7. Real component: arctan(z)
    8. Imaginary component: arctan(z)
    9. Real component: log z
    10. Imaginary component: log z
  5. Integration
    1. Parametrization of curves in the complex plane
    2. Tangent parametrization
    3. Circle in the complex plane
    4. Circle orientation
    5. Complex integration: (z-1)(conj(z)^2+conj(z)+1)
    6. Complex integration: z^(1/2)
    7. Complex integration: z^i
    8. Complex integration: conj(z)
    9. Complex integration: Antiderivatives
    10. Exploring roots
    11. Visualization of the Fundametal Theorem of Algebra
    12. Cauchy Integral Formula: Example 2
    13. Cauchy-Goursat theorem: Example
  6. Series & Taylor series
    1. Geometric interpretation of sequences
    2. Geometric series complex numbers
    3. Taylor series
    4. Mandelbrot iteration orbits
  7. Conformal mapping
    1. Conformal mapping: Visualization
    2. Exploring conformal mapping
    3. Mappings of the circle to the upper/lower half plane

1.

Preliminaries

  • 1. Complex numbers: cartesian representation
  • 2. Complex numbers: Polar and exponential forms
  • 3. Complex arithmetic: geometric interpretation
  • 4. Complex Roots
  • 5. Argument and Principal argument
  • 6. Multiplication of complex numbers as stretching (squeezing) and rotation
  • 7. Polar multiplication of complex numbers
  • 8. Addition-Subtraction of complex numbers
  • 9. Explore quadratic equations
  • 10. Explore cubic equation

1.1.

Complex numbers: cartesian representation

– Juan Carlos Ponce Campuzano

1.2.

–

1.3.

–

1.4.

–

1.5.

–

1.6.

–

1.7.

–

1.8.

–

1.9.

–

1.10.

–

2.

Functions

  • 1. Real and Imaginary components
  • 2. Riemann sphere
  • 3. Complex differentiation

2.1.

Real and Imaginary components

– Juan Carlos Ponce Campuzano

2.2.

–

2.3.

–

3.

Mappings

  • 1. Mappings
  • 2. Inversion 1/z
  • 3. The mapping 1/z
  • 4. The Joukowsky map
  • 5. Complex mappings

3.1.

Mappings

– Juan Carlos Ponce Campuzano

3.2.

–

3.3.

–

3.4.

–

3.5.

–

4.

Riemann surfaces: Examples

  • 1. Real component: z^(1/2)
  • 2. Imaginary component: z^(1/2)
  • 3. Real component: z^(1/3)
  • 4. Imaginary component: z^(1/3)
  • 5. Real component: (1-z^2)^(1/2)
  • 6. Imaginary component: (1-z^2)^(1/2)
  • 7. Real component: arctan(z)
  • 8. Imaginary component: arctan(z)
  • 9. Real component: log z
  • 10. Imaginary component: log z

4.1.

Real component: z^(1/2)

– Juan Carlos Ponce Campuzano

4.2.

–

4.3.

–

4.4.

–

4.5.

–

4.6.

–

4.7.

–

4.8.

–

4.9.

–

4.10.

–

5.

Integration

  • 1. Parametrization of curves in the complex plane
  • 2. Tangent parametrization
  • 3. Circle in the complex plane
  • 4. Circle orientation
  • 5. Complex integration: (z-1)(conj(z)^2+conj(z)+1)
  • 6. Complex integration: z^(1/2)
  • 7. Complex integration: z^i
  • 8. Complex integration: conj(z)
  • 9. Complex integration: Antiderivatives
  • 10. Exploring roots
  • 11. Visualization of the Fundametal Theorem of Algebra
  • 12. Cauchy Integral Formula: Example 2
  • 13. Cauchy-Goursat theorem: Example

5.1.

Parametrization of curves in the complex plane

– Juan Carlos Ponce Campuzano

5.2.

–

5.3.

–

5.4.

–

5.5.

–

5.6.

–

5.7.

–

5.8.

–

5.9.

–

5.10.

–

5.11.

–

5.12.

–

5.13.

–

6.

Series & Taylor series

  • 1. Geometric interpretation of sequences
  • 2. Geometric series complex numbers
  • 3. Taylor series
  • 4. Mandelbrot iteration orbits

6.1.

Geometric interpretation of sequences

– Juan Carlos Ponce Campuzano

6.2.

–

6.3.

–

6.4.

–

7.

Conformal mapping

  • 1. Conformal mapping: Visualization
  • 2. Exploring conformal mapping
  • 3. Mappings of the circle to the upper/lower half plane

7.1.

Conformal mapping: Visualization

Support this project
https://www.patreon.com/jcponce[br]
– Juan Carlos Ponce Campuzano

7.2.

–

7.3.

–

Information