Understanding Complex Variables and Derivation of the Red Box Explained

[member 428835]

Hi PF!

Attached is a very small piece of my professor's notes. I would write it out here but you need the picture. Referencing that, $z$ and $\zeta$ are complex variables, the real components listed on the axes, where the vertical axis should have the imaginary $i$ next to it. What I don't understand is how he arrives at the red box. From what I can understand, looking at the second $z$ and $\zeta$ definitions and substituting one in for the other through the variable $r$ we have $$z = \zeta e^{-i \pi+i\theta_0}$$ but from here I don't see how to arrive at the boxed in piece. Any help would be awesome!

I should say I searched everywhere online but no one showed the derivation.

 

[FactChecker]

For the natural number n>0, zⁿ is a 1-1 conformal mapping from the area above the angle, Θ₀ = π/n, on the left to the upper half plane on the right. That will give you a way of mapping the straight horizontal lines on the right upper half plane to the flow lines of an incompressable, irrotational flow on the left within the angle area.