# Conformal mapping with pgfplots

I would like to realize the following graph with pgfplots:

which represents the following conformal mapping:

u=x^{1/q}
v=y^{1/q}

where q=2*(pi-alpha)/pi.

EDIT:

Here is the formula for the conformal mapping, as reported in the original article:

I really don't know from where should I start to realize a graph of this kind, so every suggestion is welcome. Once I have some useful suggestion, I will try to write my own code.

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What you want to achieve is a visualization of the complex function f(z) = u(z) + i v(z) with z = x + i y.

In your image, you see two contour plots in the same axis: one for u(z) and one for v(z).

There are only two questions remaining:

1. how can you plot two contour plots of given functions into the same axis, and how can you control their appearance?

2. what are the formulas for u(z) and v(z)?

I can assist you with 1. Concerning 2., I have not been able to reproduce you graph by means of your provided functions. Can you verify them? Are you sure that u(z) = u(x) does not depend on y? And, similarly, that v(z) = v(y) does not depend on x? You image seems to indicate that u and v depend on both x and y. And what is alpha? It seems to be something like pi/5 (that's what I guessed).

So, here is a solution for question (1). Suppose we want to visualize f(z) = z^2 = (x+iy)^2 = (x^2 - y^2) + xy i :

\documentclass{article}

\usepackage{pgfplots}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
view={0}{90},
xlabel=real axis,
ylabel=imaginary axis,
]