I haven't taken complex analysis yet so there may be many words to sift through and not much concrete mathematical notation. I'll try my best though.
Given a local Euclidean unit square grid how would I go about finding a suitable conformal mapping function that transforms the real and imaginary grid lines to match those as pictured in the right-most square?
Do I need two separate transformation functions, one to transform the real gridlines and another to transform the imaginary gridlines? Would this approach work?
Algebraic-topologically speaking there is an equivalence relation imposed upon the left and right side of the square, and similarly there is an equivalence relation imposed upon the top and bottom of the square, such that all gridpoints incident upon the left side of the square are all "mapped" or "equated" with the top left corner of the square. The same thing is going on for the imaginary gridlines. There are multiple ways to do this, but the transformation seems to rotate each gridline and stretch them too.
So it seems like the transformation function would be a "multi-step operation" such that first, the equation of points incident upon the left side of the square would be made, the equation of points incident upon the right side of the square would be made, and then the whole structure would be rotated 45 degrees clockwise, and then stretched to the corner points opposite via the diagonal of the square. The same sort of description would apply for the imaginary curves. It's a bit too intricate of a transformation for me to figure out the exact function. I'm stuck. If anyone has ideas on how to proceed from here I welcome them.
This is an outline of what I would do:
First.- In the left graph, the red vertical lines are defined as $$x=k$$ with k=[0,1]
Second.- Apply a linear transformation $$x1=k1$$ with k1=[0,$\pi$/2]
Third.- Apply the tangent function $$x2=tan(x1)$$ with k2=[0,infinite)
Fourth.- Apply $$v=u^{(x2)}$$ with u=[0,1] and with v=[0,1]