source/reference:
https://www.youtube.com/channel/UCaTLkDn9_1Wy5TRYfVULYUw/playlists

Complex Function

Function

In general, a function ƒ:A→B is a rule that assigns to each element of A exactly one element of B. for example, ƒ:ℝ→ℝ, ƒ(x)=x²+1. Since ℝ can be represented by a one dimensional number line, the function mapping can be represented by a graph in the x-y plane.

Complex Function

For a complex function, for example, ƒ:ℂ→ℂ ƒ(z)=z²+1.

Let z=x+𝑖y, w=ƒ(z)
⇒w=ƒ(z)=(x+𝑖y)²+1
⇒w=(x²-y²+1)+𝑖(2xy)
⇒w=u(x,y)+𝑖v(x,y) where u,v:ℝ²→ℝ

Graphing Complex Function

Two complex planes are used to graph the complex function. One is for the domain and one is for the range. These complex planes can be used to analyze how geometric configurations in the z-plane are mapped under ƒ to the w-plane.

For example

Let w=ƒ(z)=z² , z=re𝑖θ
⇒w=(re𝑖θ)²=r²e2𝑖θ
⇒|w|=|z|², arg w=2arg z

As z moves around a circle of radius r in z-plane once, w moves around the circle of radius r² at double speed twice in w-plane.

For example

Let w=w'+c, c∊ℂ, w'=ƒ(z)=z²,  z=re𝑖θ
⇒w'=(re𝑖θ)²=r²e2𝑖θ
⇒|w'|=|z|², arg w'=2arg z
∴w=r²e2𝑖θ+c