I’m looking for any example of a conformal map m: P -> Q where P is some polygon of at least 4 sides, Q is a rectangle, and and m is not linear (so P and Q are not merely scaled, rotated, & translated copies of each other). FWIW, Q must in general depend on P, which is why I’m not specifying Q further here.
- I don’t care how many sides P has for this example — I want it in order to validate some more general work. (Especially helpful if the number of sides is “small”, say < 7 or even 10. Even better if it has 4 sides (not all right angles), but that’s not necessary.)
- I understand that m won’t be unique. I’m just looking for ANY such m.
- Crucially, I need m to be exact, symbolic, constructed, and fully “worked out”. By that, I mean no limits, no implicit functions (or ones with only an existence proof but no construction), no indefinite unevaluated integrals, etc. I am flexible in what class of functions is used to express m, but think of those in, say, Abramowitz & Stegun, or implemented in the core of MATLAB. E.g., elementary, incomplete beta, Bessel, and elliptic-integral functions are all fine.
The reason I phrase #3 that way is that I want to be able to inspect m, look at its derivatives, etc., which will be opaque if the function is not explicitly written in terms of such functions. What makes the conditions of #3 useful for that is that those functions (a) satisfy various analytic and algebraic relations, with published theorems behind them, and (b) have already been implemented in code according to established standards of accuracy and numerical stability (something which might not automatically be true of their inverses, BTW).