Complex Analysis

Juan Carlos Ponce Campuzano, Mar 2, 2015

A visual point of view of different topics related to Complex Analysis

Table of Contents

  1. Complex numbers
    1. Roots of a complex number
    2. Basic operations with complex numbers
    3. Complex roots
  2. Mappings
    1. The function f(z)=1/z (Part 1)
    2. The function f(z)=1/z (Part 2)
    3. The function f(z)=z/|z|^2
    4. The function f(z)=sin(z) (Part 1)
    5. The function f(z)=sin(z) (Part 2)
    6. The function f(z)=exp(z)
    7. The function f(z)=log(z) (Part 1)
    8. The function f(z)=log(z) (Part 2)
    9. The function f(z)=log(z+1)
    10. Möbius transformation
    11. Mapping squares
    12. Mapping squares
    13. Mapping squares
    14. Complex mappings
    15. Complex mappings
  3. Riemann Surfaces
    1. Riemann Surface: z^(1/2)
    2. Riemann Surface: log(z)
    3. Riemann Surface: z^(1/3)
    4. Riemann Surface: (1-z^2)^(1/2)
    5. Riemann Surface: arctan(z)
    6. Riemann surface: f(z)= arg(z)
  4. Analytic functions
    1. Riemann sphere
    2. Riemann sphere (2nd part)
    3. Visualising zeros of complex polynomials
    4. Graphs of complex functions
    5. Complex functions
    6. Complex functions: Domain coloring
    7. Complex functions: Domain colouring with contour lines
  5. Series
    1. Series (Part I)
    2. Series (Part II)
    3. The exponential complex function
    4. Complex series
    5. Complex spiral
    6. Complex spiral
  6. Applications
    1. Hydrodynamics: Streamlines and velocity potentials
    2. Complex potential

1.

Complex numbers

  • 1. Roots of a complex number
  • 2. Basic operations with complex numbers
  • 3. Complex roots

1.1.

Roots of a complex number

To solve the equation [math]z^n=a+bi[/math] for [math]a,b\in \mathbb R[/math] and [math]n\in \mathbb N[/math]
Change the values of a, b, and n.
– Juan Carlos Ponce Campuzano

1.2.

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1.3.

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2.

Mappings

Mappings by different functions

  • 1. The function f(z)=1/z (Part 1)
  • 2. The function f(z)=1/z (Part 2)
  • 3. The function f(z)=z/|z|^2
  • 4. The function f(z)=sin(z) (Part 1)
  • 5. The function f(z)=sin(z) (Part 2)
  • 6. The function f(z)=exp(z)
  • 7. The function f(z)=log(z) (Part 1)
  • 8. The function f(z)=log(z) (Part 2)
  • 9. The function f(z)=log(z+1)
  • 10. Möbius transformation
  • 11. Mapping squares
  • 12. Mapping squares
  • 13. Mapping squares
  • 14. Complex mappings
  • 15. Complex mappings

2.1.

The function f(z)=1/z (Part 1)

Mapping lines and circles with the function [math]f(z)=\frac{1}{z}[/math]
The function f(z)=1/z (Part 1)
– Juan Carlos Ponce Campuzano

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2.7.

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2.10.

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2.11.

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2.12.

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2.13.

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2.15.

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3.

Riemann Surfaces

  • 1. Riemann Surface: z^(1/2)
  • 2. Riemann Surface: log(z)
  • 3. Riemann Surface: z^(1/3)
  • 4. Riemann Surface: (1-z^2)^(1/2)
  • 5. Riemann Surface: arctan(z)
  • 6. Riemann surface: f(z)= arg(z)

3.1.

Riemann Surface: z^(1/2)

GGB script
Re1(x, y) = exp(1 / 2 * log( sqrt( x*x + y*y ) ) ) * cos(1 / 2 * arctan2( y, x ))[br]Im1(x, y) = exp(1 / 2 * log( sqrt( x*x + y*y ) ) ) * sin(1 / 2 * arctan2( y, x ))[br][br]t = Slider(0, 1, 0.1, 1, 100, false, true, false, false)[br][br]HRe1(x, y) = 0 * (1-t) + Re1(x, y) * t[br]HIm1(x, y) = 0 * (1-t) + Im1(x, y) * t[br][br]Re2(x, y) = exp(1 / 2 * log( sqrt( x * x + y*y ) ) ) * cos(1 / 2 * (arctan2( y, x ) + 2 pi))[br]Im2(x, y) = exp(1 / 2 * log( sqrt( x * x + y*y ) ) ) * sin(1 / 2 * (arctan2( y, x )+2 pi))[br][br]HRe2(x, y) = 0 * (1-t) + Re2(x, y) * t[br]HIm2(x, y) = 0 * (1-t) + Im2(x, y) * t[br][br]RSRe = Surface(u*cos(v), u*sin(v), HRe1(u*cos(v), u*sin(v)), u, 0, 2, v, -pi, pi-pi/200)[br]RSIm = Surface(u*cos(v), u*sin(v), HIm1(u*cos(v), u*sin(v)), u,0, 2, v, -pi, pi-pi/200)[br]RSReN = Surface(u*cos(v), u*sin(v), HRe2(u*cos(v), u*sin(v)), u, 0, 2, v, -pi, pi-pi/200)[br]RSImN = Surface(u*cos(v), u*sin(v), HIm2(u*cos(v), u*sin(v)), u, 0, 2, v, -pi, pi-pi/200)
– Juan Carlos Ponce Campuzano

3.2.

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3.3.

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3.5.

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3.6.

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4.

Analytic functions

  • 1. Riemann sphere
  • 2. Riemann sphere (2nd part)
  • 3. Visualising zeros of complex polynomials
  • 4. Graphs of complex functions
  • 5. Complex functions
  • 6. Complex functions: Domain coloring
  • 7. Complex functions: Domain colouring with contour lines

4.1.

Riemann sphere

The extended complex plane can be mapped onto the surface of a sphere whose south [br]pole corresponds to the origin and whose north pole to the point [math]\infty[/math].
Move the slider x and y to move the point [math]z=x+iy[/math]
– Juan Carlos Ponce Campuzano

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4.7.

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5.

Series

  • 1. Series (Part I)
  • 2. Series (Part II)
  • 3. The exponential complex function
  • 4. Complex series
  • 5. Complex spiral
  • 6. Complex spiral

5.1.

Series (Part I)

An infinite series [math]\sum_{n=1}^{\infty}z_n[/math] of complex numbers converges to the sum [math]S[/math] if the sequence [math]S_N=\sum_{n=1}^{N}z_n[/math] of partial sums converges to [math]S[/math]. We then write [math]\sum_{n=1}^{\infty}z_n=S[/math]. For example, [math]\sum_{n=1}^{\infty}z^n=\frac{1}{1-z}[/math].[br][br]Drag the point [math]z[/math] around to see what happens to the partial sums when [math]z[/math] is inside or out of the open ball [math]|z|=1[/math].
Series (Part I)
– Juan Carlos Ponce Campuzano

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6.

Applications

  • 1. Hydrodynamics: Streamlines and velocity potentials
  • 2. Complex potential

6.1.

Hydrodynamics: Streamlines and velocity potentials

If we have a (steady-state) incompressible, nonviscous fluid, we are interested in finding its velocity field [math]V(x,y).[/math] [br]If [math]F=\phi+i\psi[/math] is the complex potential, the functions [math]\phi[/math] and [math]\psi[/math] are the [b]velocity potential[/b] and [b]stream[/b] functions, respectively. In this case [math]V=\text{grad}\,\phi[/math].
Hydrodynamics: Streamlines and velocity potentials
– Juan Carlos Ponce Campuzano

6.2.

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