My question refers to some not very well known fragments of Gauss that treat the problem of finding a conformal mapping (angle-preserving mapping) in the complex plane from the interior of the ellipse to the interior of the unit circle. These fragments date from 1839, much later then his better-known article from 1822 on conformal mappings. I guess the mathematics involved here is connected somehow with the later Riemann's mapping theorem, which states that every simply-connected subset of the complex plane is "conformally equivalent" to the interior of unit circle.

The relevant pages from Gauss's nachlass are in volume 10-1, p. 311-320. Gauss's formulas in this fragment are interesting since they seem to be very complex - this stands in contrast to the reltively simple solution to this problem using the "Joukowsky transform" (as can be seen from this post: I even found (in the last pages) some formulas that look like theta functions. It's noteworthy that in his essay, Schlesinger comments on this fragment of Gauss at p. 192, so i think it might be of importance (i didn't find comments about it the literature).

So just to be clear, i ask this question simply because i don't understand the mathematical framework of this problem. I think that unlike other questions i asked before (like the question on Gauss's "conception of braids"), this question can be answered by a mathematician who is not an historian, but has enough knowledge in the theory of conformal mappings. So i hope for a good answer!

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