# Explanation of Gauss's late fragments dealing with “the conformal image of the ellipse”

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: https://math.stackexchange.com/questions/1582608/conformally-mapping-an-ellipse-into-the-unit-circle). 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!

According to p.192-194 of Schlesinger's essay, Gauss was led to the problem of conformal mapping of the interior of the ellipse into the interior of the unit circle around the year 1834. His first attempt, from 1834, was futile, since he was unable to find the mathematical expression of the mapping.

His second attempt, dating from 1839, was successful, as he was ocuppied at this period with the (related) theory of the potential in physics and mathematics and composed his treatise "General propositions on inverse-squared forces" (published in 1840). He succeeded in constructing the so-called Green's function of ellipse with the centre at the pole. Gauss then expressed the resulting conformal mapping using his theory of theta series and theorems from the theory of potentials (Schlesinger mentions article 29 from "general propositions" as connected with this problem).

Schlesinger also mentions that Gauss's result matches the formula of Hermann amadeus Schwarz from 1870 (although one also has to use a formula of Jacobi in order to make the step from Gauss's to Schwarz's formula).

• I added this as a partial answer - anyone with additional useful comments will be welcomed! – user2554 Nov 16 '18 at 16:33