How to Visualize z^2, 1/z, & e^z as Complex Mappings of the Complex Plane (use Wolfram Mathematica)

• Van Aubel's Theorem ha... .. Complex-valued functions of a complex variable can be visualized in many ways. We can graph their real and imaginary parts individually. Or, we can graph the function as a "whole" by considering it to be a mapping (transformation) of the complex plane to itself. The examples we consider in this video are: 1) w=f(z)=z^2 (squaring map), 2) w=f(z)=1/z (reciprocal map), and 3) w=f(z)=e^z (exponential map). Where derivatives are nonzero, these mappings are conformal transformations (they preserve angles). Of these examples, the only one that has a derivative equal to zero is example 1 (f'(0)=0). The amplitwist concept is helpful to interpret derivative values. We use Wolfram Mathematica to help us do these visualizations.
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(0:00) Complex variables
(0:37) w=f(z)=z^2 (Squaring mapping)
(4:40) Expand f(x+iy) to find u(x,y) and v(x,y) (real and imaginary parts)
(7:51) Corresponding transformation Φ(x,y)=(u(x,y),v(x,y)) of the xy-plane to the uv-plane
(9:46) Mathematica exploration
(13:45) Orthogonal level curves ("trajectories") of real and imaginary parts (conformal mapping)
(15:38) Vector field
(16:10) Mapping animation
(18:59) w=f(z)=1/z (Inverting mapping), find u(x,y) & v(x,y) (real and imaginary parts)
(21:45) Mathematica exploration
(23:44) Orthogonal level curves
(24:23) Mapping animation
(27:16) w=f(z)=e^z (Exponential mapping), find u(x,y) & v(x,y) with Euler's identity
(30:56) Mathematica exploration
(32:11) Orthogonal level curves & mapping animation
(33:18) Derivatives and the amplitwist property
(34:43) Integration and the Residue Theorem
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[@megha3336]

Nice explanation

[@billkinneymath]

Thanks! Glad you liked it!