Conformal mapping with pgfplots

Question

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.

Thanks for your kind help.

Answer 1

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,
        ]
    \addplot3[contour gnuplot={number=9,labels=false,draw color=blue}] {x^2-y^2};
    \addplot3[contour gnuplot={number=9,labels=false,draw color=red}] {x*y};
    \end{axis}
\end{tikzpicture}
\end{document}

Here, I used draw color to change the color (which is typically mapped color for contour plots).