# Meaning of passages by Gauss on the “convergence of expansions (in infinite series) of the (elliptical) equation of the center”?

Yesterday I took my time to look again into Schlesinger's essay on Gauss's contributions to analysis, and I found something new I didn't know about (so it caught my eye) in the last subsection of the chapter "The posthumous treatise on the agM [apparently 'arithmetico-geometric Mean']. Extension to the theory of module function" (p. 84-117). In subsection (e) of this chapter, named "The last diary notes from 1800. Asymptotic. Midpoint equation. Class number", Schlesinger comments on the equation of the center, includes some intriguing results of Gauss, and mentions that his formulas agree with that of Jacobi (1849). The relevant pages in Gauss's nachlass are p. 420-428 of volume 10-1 of his works, and they are entitled "on the convergence of expansions [in series] of the equation of the center" (German: 'Über die Konvergenz der Entwicklung der Mittelpunktsgleichung'). It seems that these pages are actually part [VII] of a planned treatise on the convergence of infinite series.

Just to be clear, I know nothing about this so called "Mittelpunktsgleichung", and I'm not even sure if this work was done by Gauss in astronomical contexts or analytic contexts, or perhaps both (I mention astronomical context since it seems that these formulas were connected with his work on "Kepler's equation" in astronomy).

So I'm not looking for a very sophisticated answer, just an explanation of what Gauss actually did and the context it was done. I think that if Schlesinger commented on it then it's important.

• I've tried to clarify the question by using recognized English terms that match the original German so far as that is accessible, and giving original German title(s) again so far as identifiable, to help locate sources. It would probably help understanding of the question and production of answers if questioner could give a complete original citation for the other titles that I couldn't identify. Of course if I haven't succeeded in matching the intent of the question, please revert or name the problem. – terry-s Mar 26 '19 at 2:05

The "midpoint equation" is another name for the "equation of the centre", related to kepler's equation. What Gauss wanted to determine in this investigation is the exact mathematical expression for the coefficients of the trigonometric series (Fourier series) for the difference between true anomaly $$v$$ and mean anomaly $$M$$:

$$v - M = \sum_{n=1}^\infty \frac{{1}}{{n}}C_n \sin(nM)$$

Gauss's formula for the coefficients is: $$C_n = (\tan(\frac{{\phi}}{{2}})e^{\cos\phi})^n(1 + \frac{{4}}{{3}}\frac{{1}}{{\sqrt{{2n\pi \cos^3\phi}}}})$$, where $$e$$ is the base of natural logs and $$\phi$$ is a the arcsine of the eccentricity $$\varepsilon$$ ($$\varepsilon = \sin\phi$$).

It seems that the significance of this investigation of Gauss lies in the fact that his method of derivation is based on his ideas on the foundations of complex analysis - in order to derive the closed form expression for the coefficients of the series he applied a cetain imaginary transformation for the eccentric anomaly $$E$$ (the imaginary transformation is $$E = i\cdot \log\cot(\frac{{\phi}}{{2}}) + \epsilon$$ ). Gauss doesn't expain how he arrived at this method, as he is operating entirely with series representations, but later the mathematician Jakob Horn, who was known for his contributions to the theory of asymptotic expansions, showed that one can arrive at this susbtitution by operating with integrals in the complex plane and using Cauchy's residues theorem.

In his historic reconstruction of Gauss's line of thought, Schlesinger also mentiones some preliminiary research done by Gauss and recorded in several diary entries. These entries are concerned with summation of divergent series (much in the Euler's tradition) and the asymptotic behaviour of Bessel functions and the location of their zeroes (see Gauss's werke, volume 10, p.382-389).

In the last years of his life, Gauss attempted to write a mature treatise on convergence of series which will summarize his lifetime occupation with series theory. Since Gauss expected great future for the (still infant) subject of analysis situs (topology), one might speculate that Gauss wanted to combine his ideas on behaviour of complex functions around singularities (he knew Cauchy integral theorem) and topological ideas about connectivity of surfaces. I think that the article "What Gauss told Riemann about Abel’s Theorem" is related to this historic thesis.

A little bit of history:

Very late in his life (1850 and on) Gauss mentioned in several letters to his friend Schumacher his wish to write a fundamental treatise on the convergence of trigonometric series, which will include the connections to the doctrine of complex quantities . He claims priority for Jacobi's 1849 researches on kepler's equation, stating that he possesed the solution to the problem of coefficients of the trigonometric series for about $$\pm 50$$ years. The exact year at which he made his relevant researches cannot be dated exactly, but Schlesinger speculates that it was around the year 1805.

Actually the first mathematical astronomer who tried to solved this problem was Carlini, who gave a flawed solution in 1817, so Gauss's solution predates even him.

The significance of Gauss's way of deriving the solution is that it's incomparably shorter than that of Jacobi. If i'm not mistaken, one of the possible subjects Gauss suggested to Riemann for his doctoral dissertation was trigonometric series, so i suspect that (since it was around the same year of his letters to schumacher) it might be connected somehow with that, but this is entirely a speculation of mine.

• Thanks for this answer. I didn't previously know of Gauss's interesting closed-form expression for coefficients $C_n$ in the elliptical eqn of the center, but also couldn't find it in the Schlesinger article or Gauss' Werke 10-1 as you cited. Could you quote a citation for this $C_n$ result and $E$ transformation of Gauss, plus Carlini's 1817 work? Also it would be helpful if you could clarify the symbol usage? perhaps $\epsilon$ is Gauss' eccentricity and $e$ the base of exponentials and natural logs? Is $\phi$ defined as it often was, the arcsine of the eccentricity? Thanks in advance. – terry-s Mar 25 '19 at 13:58
• Thanks. I've checked it again and it seems that $\epsilon$ is really the eccentricity, e is the base of the natural logarithm, and $\phi = arcsin \epsilon$ (Gauss defines it so in the last sentence of p.420). For references in Schlesinger article or Gauss' werke, see p.438-447 in volume 10-1 and p.115-116 in Schlesinger article. The first reference i gave is much more comprehensive and includes a reconstruction (by J. Horn) of the idea (on p.446) behind the imaginary substitution he employs. As for Carlini's 1817 work - i don't know where is it; i just cited what Schlesinger said. – user2554 Mar 25 '19 at 15:19
• Sorry if i'm not helping a lot; the mathematics here is difficult and i also always have to use Google translate from german to english, and that adds additional difficulty. Meanwhile, i'll correct the symbol usage in my answer. – user2554 Mar 25 '19 at 15:22
• @terry-s By the way, you can also look at p.210-212 of the book "An introduction to the Mathematics and Methods of Astrodynamics" - the author (Richard H. Battin) gives a developement of the center equation in a fourier series by means of complex analytic methods - by integration along a circular contour and evaluation of these integrals using Cauchy's residues theorem. I don't know if the method in the book is the same as Gauss's, but apparently Gauss was the first to use integrals along contours in the complex plane in order to solve a real-valued problem. – user2554 Mar 25 '19 at 19:17
• After looking at the Gauss source and the commentary on page 441 of vol.10-1, it seems clear that te expression for $C_n$ is not an exact expression for the coefficient, it is an asymptotic approximation that gets closer for large values of n. As such, the whole purpose of that form of expression seems to be to investigate convergence by giving a close-enough equivalent to the original term, in a form that can be subjected to standard convergence tests. But the given form of $C_n$ would apparently not be useful for calculating values of the series. .../... – terry-s Mar 26 '19 at 2:18