# How to interpret Gauss's late fragments on conformal mapping of the interior of an ellipse (to the unit disk) in modern mathematical terms?

My question refers to some not very well known (and unpublished) fragments of Gauss that treat the problem of finding a conformal mapping (angle-preserving mapping) in the complex plane from the interior of the ellipse to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. The relevant pages from Gauss's nachlass are in Werke volume 10-1, p. 311-320 (archive.org). Schlesinger comments on these fragments of Gauss on p.192 of his essay Über Gauss's Arbeiten zur Funktionentheorie (archive.org), where he mentions that his results match the formula found much later (1870) by Hermann Schwarz in a short notice (archive.org) in which he uses the recent constructive methods (methods which Schwarz played a central role in their developement) of building a conformal mapping from simply connected regions to the unit disk. Schwarz himself was encouraged to develope a vast amount of such techniques by his mentor Weierstrass, who wasn't satisfied by Riemann's existence proof of his famous mapping theorem.

Gauss's formula:

Gauss refers to an ellipse in the $$T$$ complex plane ($$T=x+yi$$) whose equation is $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$. Rewriting Gauss's results with Jacobi's notation, the function that mapps conformaly this ellipse in the $$T$$ plane to a unit circle in the $$t$$ plane is:

$$t= \sqrt{k}\cdot \mathbb{sin \space am} (\frac{2K}{\pi}\mathbb{arcsin}(\frac{T}{2\mu}))$$

where $$k = \frac{b^2}{a^2}$$, $$\mathbb{sin \space am}$$ is Jacobi's "Sinus Amplitude" function ,$$K$$ is the complete elliptic integral of the first kind of modulus $$k$$, and $$\mu$$ is derived from the "theta function" relation: $$a=1+2\mu^4+2\mu^{16}+2\mu^{36}+...$$.

Some details on Gauss's solution process

It seems that Gauss discovered this solution by a combination of his early ideas on elliptic integrals (and theta functions) and his more mature ideas on potential theory - in the process of solution, Gauss describes a certain physical construction which reflects the deepening of his understanding of the "logarithmic potential". Gauss refers to a two-dimensional potential problem - in which forces decay according to $$\frac{1}{r}$$ (and not to $$\frac{1}{r^2}$$ like in Newtonian three dimensional gravity) and potentials according to $$\mathbb{log}(r)$$ - and derives the mass density distribution of an elliptical ring that satisfies the following condition: the potential on the periphery of the ellipse (on the elliptical ring) must coincide with that of a point mass with mass equal to the mass of the ring (Gauss denotes the mass by $$2\pi A$$) which is located at the center of the ellipse.

If the ring was a circle than obviously the required mass distribution would be uniform; but the ring is elliptical and hence the great difficulty. Unfortunately i wasn't able to infer from Gauss's fragments and Schlesinger's comments how he translated this pure mathematics problem into the physical problem. Schlesinger says that Gauss's manner of solution of this potential-theoretic problem involves the construction of the Green's function of an ellipse with the center at the pole, and that the solution apparently enabled him to derive the conformal mapping of the ellipse interior.

Significance of Gauss's findings fron an historical perspective

These fragments are very noteworthy, first of all because they seem to anticipate the Riemann's mapping theorem (why did he attempt to find such a solution at all?), and secondly because they introduce powerful mathematical tools for solving the problem of explicit construction of conformal mapppings to the unit circle (this problem seems to be even more complex than the so-called Schwarz-Christoffel maps for mappings of polygonal interiors by elliptic integrals).

In addition, it must be remarked that while it's well known that Gauss coined the term "potential" in his published work in 1840 (actually George Green first coined the term in 1828, but his treatise wasn't very well known), it's much less known that the first appearence of the term "potential" in Gauss's writings was in those fragments (this fact was pointed out by Clemens Schaefer on p.95 of his essay Über Gauss's Physikalische Arbeiten (Magnetismus, Electrodynamik, Optik)).

I've already added a very partial answer at HSM stack exchange, of which i'm not very satisfied. According to several articles i found, this is a very dificult problem and surprisingly i didn't find any comment on Gauss's solution to it in the literature (except Schlesinger's comment). So i'll be glad if anyone will explain what is going on there in Gauss's writings from a modern viewpoint.

Update (29.06.23)

I found an additional reference to Gauss's treatment of this problem in the book "Dirichlet's principle : a mathematical comedy of errors and its influence on the development of analysis" (1975) by Monna, A. F. Subsection 1 of of chapter 2 is dedicated to Gauss's contributions to potential theory, and on p.11-12 the following is written:

The sources of Gauss's work on potential theory are in three fields. First the problem of the attraction of the ellipsoid on a point according to Newton's law. The corresponding problem for a sphere was already solved by Newton. The problem for ellipsoid is much more difficult... Gauss treated this problem in a paper from 1813... In the second place Gauss was motivated by the theory of conformal mappings, in particular by the problem of the conformal mapping of the interior of an ellipse onto that of the circle. Gauss solved this problem in the years 1834-1839. It belongs to the domain of complex function theory because the mapping is defined by a complex analytic function. This problem is connected with two-dimensional potential theory and the solution of the equation $$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0$$, since, as a consequences of the differential equations of Cauchy-Riemann - the real and imaginary parts of an analytic function satisfy this equation. The problem of the existence of the mapping is therefore related to fundamental existence theorems in potential theory. In this context Gauss used the logarithmic potential mentioned before... In the third place Gauss studied potential theory in connection with his research on terrestrial magnetism...

I cited this book just because references in the literature to Gauss's solution of this problem are scarce; however, although I found in this book many interesting details on the other two fields which inspired Gauss's potential theory, the author does not enter into the details of Gauss's solution in this case. So this citation does not really help in the discussion (it just adds another perspective, by another author).

• I guess this is the HSM question you are referring to, right? hsm.stackexchange.com/q/7931/1697 Commented Jun 22, 2020 at 20:44
• @Carlo Beenakker - Yes. Commented Jun 22, 2020 at 21:02
• Is this the fragment in question: archive.org/details/werkenachtraegez101cfga/page/n313/mode/…? Commented Nov 14, 2020 at 12:22
• Also, what's the title of Schlesinger's essay, and do you have a link to it? Commented Nov 14, 2020 at 12:27
• @DavidRoberts - yes, this is the fragment in question. The title of Schlesinger's essay is "Uber Gauss's Arbeiten zur Funktionentheorie". Here is a link to volume 10-2 of Gauss's werke, which contains Schlesinger's essay: gdz.sub.uni-goettingen.de/id/PPN236019856. This essay is 210 pages long and what you are looking for is under the subtitles: "Die jahre 1827 bis 1843.Konforme Abbildung der Ellipse. Pentagramma mirificum". In addition, Schlesinger also comments on Gauss's solution on p.323-325 in internet archive reference you gave. Commented Nov 14, 2020 at 13:22

## 1 Answer

I'm not exacly sure whether this is what Gauss does, but the relation of the conformal map and Green function is standard. Suppose you know the Green's function for the domain with pole at $$u$$; this is a harmonic function $$G$$ that maps the boundary to zero and behaves like $$-\log|z-u|$$ at $$u$$. Consider the holomorphic function $$F(z)=G(z)+i\tilde{G}(z)$$, where $$\tilde{G}$$ is the complex conjugate of $$G$$. This function has additive monogromy of $$2\pi i$$ when going around the origin. The inverse function $$F^{-1}$$ is a $$2\pi i$$-periodic holomorphic covering map from the half-plane $${\Re z>0}$$ to the domain punctured at $$u$$. We may consider a similar map to the unit disc punctured at the origin, given by $$\exp(-z)$$. Hence, $$z\mapsto\exp(-F(z))$$ is the desired conformal map.

To find the Green's function, one needs to solve Dirichlet problem - find a harmonic function $$h$$ equal to $$-\log|z-u|$$ on the boundary; then $$-\log|z-u|-h$$ is the desired Green's function. What you are describing seems to be constructing such a function as a simple layer potential. For this, you need to somehow invert the operator that sends a mass distribution on a simple layer it its potential; I have no idea how Gauss did it.

• @Kostya_l - i apollogize for taking so much time to response; this subject (Green's function, conformal mappings, etc) is truly new to me, so i had to spend much time for reading about these matters. Although your answer didn't fully shed light on Gauss's fragments, it did help me understand the context, however (and this is why i upvoted you). I have to add that Schlesinger's analysis of Gauss's fragments seems to agree with you - Schlesinger writes: "according to today expressions, the difference between these two potentials is nothing else than the Green's function... " Commented Apr 4, 2021 at 19:59
• Schlesinger also writes: "In art [6], Gauss forms the difference of the complete potentials: he imagines that each of the two potentials is supplemented by adding the so-called conjugate function to a monogenic function of the complex variable $x+yi$ and substracting these two functions from one another; if this difference is set equal to $\mathbb{log} (t)$, then t evidently yields the mapping of the ellipse onto the unit circle. In particular, the series [14] is identical to [7]". The series [7] is given with the remark "the law found", together with the date of the discovery. Commented Apr 4, 2021 at 20:08
• Will it help you dechiper those fragments if i'll include in my posted question a translation of several things Schlesinger says? Commented Apr 4, 2021 at 20:16