Explanation of an imaginary transformation occurring in the determination of trigonometric series for the elliptical equation of the center

Question

One of the oldest problems in astronomy, which dates back to to the time of Kepler, is the problem of development in infinite trigonometric series of the "equation of the center" - to express the difference between true anomaly $v$ and mean anomaly $M$ as a trigonometric series of integer multiples of mean anomaly $M$, with coefficients of the series that are functions of the eccentricity $\epsilon$:

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

where $C_n = f(\epsilon)$. The problem of determination of this trigonometric series is essentially a modern equivalent of the ancient method of epicycles - this was a model of the celestial positions of the planets which used a series of "wheels" ("epicycles") with different radiuses and angular speeds to trace the location of the planet. The coefficient $\frac{C_n}{n}$ is essentially the radius of the $n$th wheel, while it's angular frequency is $\frac{nM}{t}$ ($t$ is the time of a given mean anomaly measurement $M$).

Restating the problem in a more formal way, it amounts to finding the following integral (Fourier coefficients):

$$C_n = \frac{1}{\pi} Re \left[ \int_{-\pi}^{+\pi} \left( \frac{dv}{dM} \right) e^{inM} dM \right] = \frac{1}{\pi} Re \left[ \int_{-\pi}^{+\pi} \left( \frac{dv}{dM}\cdot \frac{dM}{dE} \right) e^{inM} dE \right]$$

Since $\frac{dv}{dM}$ can be expressed through the eccentric anomaly $E$ and the eccentricity $\epsilon$ by the kepler equation, this problem is closely related to Kepler equation. Since Kepler wrote that Kepler equation cannot be solved a priori (on account of the different nature of the arc and the sine), this was historically a very challenging problem.

C.F. Gauss (unpublished, 1805), F. Carlini (in 1817) and P.S. Laplace (in 1827) solved this problem, and as a by-product discovered an interesting phenomenon that the trigonometric series diverges (it doesn't converge to $v-M$ ) for values of the eccentricities above approximately $0.662$, a value which is now called "Laplace limit". Carlini determined the trigonometric series by a very remarkable method that, according to several sources, anticipated the WKB approximation of quantum mechanics (Carlini's work was popularized by C.G.J. Jacobi in 1849), while Gauss used a different method which was apparently based on his ideas on complex analysis. Gauss, and later the mathematician Wilhelm Scheibner (Scheibner in 1856 - Ueber die asymptotischen Werthe der Coefficienten in den nach der mittleren Anomalie vorgenommenen Entwickelungen, here is a link to his article), applied a certain "imaginary transformation" connecting the eccentric anomaly $E$ and the eccentricity $\epsilon$:

$$E = i\cdot \log \cot \left(\frac{{\varphi}}{{2}} \right) + \epsilon$$

where $\varphi$ is an angle defined as $\sin(\varphi) = \epsilon$.

Now, I have many misunderstandings concerning this substitution - $E$ and $\varphi,\epsilon$ are well defined and real-sizes, so how does the imaginary unit $i = \sqrt{-1}$ enters the equation? I understand that this has something to do with contour integration in the complex plane - according to one source the Fourier integral (which I wrote before) can be transformed into a contour integral in complex variable $z$ along the circular contour $|z| = 1$ by the substitution $z = e^{iE}$. But the whole picture is very unclear to me.

According to the mathematician Jakob Horn, the meaning of this substitution can be explained in the following way:

For the developement $$C_n = \frac{1}{\pi}\int_{-\pi}^{+\pi}\frac{dv}{dM}e^{inM}dM = \frac{cos\varphi}{\pi}\int_{-\pi}^{+\pi}e^{inM}\frac{dE}{1-f cosE}$$ by introducing $z = e^{iE}$ we get $$C_n = -i\frac{cos\varphi}{\pi}\int F(z)(\Phi(z))^ndz$$ in which $$F(z) = \frac{1}{z-\frac{f}{2}z^2-\frac{f}{2}}, \Phi(z) = ze^{-\frac{f}{2}(z-\frac{1}{z})}$$ is integrated over the unit circle $|z| = 1$. The function $F(z)$ has the real singular points $$z_0 = \frac{1-\sqrt{1-f^2}}{f} = \mathbb{tan(\frac{\varphi}{2})} = e^{i\mathfrak R}<1$$ $$z_1 = \frac{1+\sqrt{1-f^2}}{f} = \mathbb{cot(\frac{\varphi}{2})} = e^{-i\mathfrak R}>1$$ which are also zeroes of $\Phi'(z)$. The integration path $$|z| = 1$$ can be replaced by the integration path $|z| = z_0$, which only has to avoid the singular point $z_0$. One has to set accordingly $z = z_0e^{i\epsilon}$, or what is the same $$E = \mathfrak R +\epsilon = i\mathbb{logcot(\frac{\varphi} {2})}+\epsilon$$ which is precisely the substitution used by Gauss. If one takes the circle $|z| = z_1$ as the integration path, bypassing the singular point $z_1$, one has $$z = z_1e^{i\theta}$$ or $$E = \theta - i\mathbb{logcot(\frac{\varphi} {2})}$$ a substitution which is used in a paper by Wilhelm Scheibner.

Despite this apparently clear explanation (and i actually did verify by hand calculation much of the facts mentioned by Horn in this passage), i really don't get the basic idea, and also have much confusion over the notation there (which involves a lot of symbols).

Therefore, my question is:

• Can anyone who is familiar with the theory of Kepler equation explain the meaning of this substitution? and how it enables to find the coefficients of the Fourier series for $v-M$?

Side remarks:

• The story of the center equation is a very convoluted one, and I'm well aware that the short background I gave in this question is not enough, so if anyone wants me to give more useful information, or add references and links, just keep me know and I'll update my posted question.
• I have already posted a more comprehensive background to this question in this HSM post, so anyone who wants more information can find it there.

Answer 1

[1] The first part of the specific question here asks what is the meaning of the 'imaginary transformation' in Scheibner's 1880 paper. If I read the paper right, there has been a confusion of symbols, perhaps partly due to Scheibner's unusual usages and habit of scattering new variables around without much explanation. The (curly) epsilon in his paper does not seem to denote the eccentricity as it often does (and as the question takes it to be): for Scheibner this symbol appears to denote just the eccentric anomaly. His 'imaginary transformation' appears in the original on p.551 (as nearly as I can reproduce it here, replacing small omega by its defined value) as --

$$\vartheta = \varepsilon + i.\mathbb{logcot(\frac{\varphi} {2})}$$

The curly theta is a complex quantity of which the real part is curly epsilon, the usual eccentric anomaly, and the imaginary part is a constant dependent on the eccentricity. A benefit of the transformation may be that curly theta does not go to zero when the real eccentric anomaly or its real functions go to zero: which may help to avoid singularities.

[2] The second part of the question asks how does this help to find the coefficients of the Fourier series for $v-M$? I believe the answer is that it doesn't do that (and it is seemingly not intended to do so).

Scheibner's (very obscure!) paper seems to have been a follow-up to efforts of better-known others, especially Carlini and Jacobi, to provide mathematically rigorous confirmation or correction of Laplace's results on convergence and divergence of series expansions that represent quantities relevant to motion in a Keplerian ellipse. Specifically, the series of interest were those expressed in terms of sines and cosines of the mean anomaly. Laplace, Carlini and Jacobi employed asymptotic approximations for distant terms of such series expansions, either adjacent each other or not far apart, to examine their contribution to convergency or divergency of the series as a whole. Scheibner's aim, not well explained, emerges from his title and his references to papers of Carlini (1817) and Jacobi (1849): he adopts a problem related to theirs, and his focus may have been to generalize, with maintenance of rigor, what had previously been done: perhaps this accounts for some of his obscurity.

The comment by @David Hammen on this question has a great deal to be said for it -- I agree with his recommendation against self-torture with such an obscure paper as Scheibner's! This is especially so as its aim and purposes are arguably now outdated and in any event of little practical use -- nobody would want to use such series at high eccentricities, even if they were technically convergent, because the convergence to practical levels of accuracy would be too slow, and would be computationally expensive, calling for too many terms.

If there is interest in trying to follow the efforts towards adding mathematical rigor to Laplace's results on the 'Laplace limit', I would be more inclined to look primarily at the papers of Laplace himself in Connaissance des Tems (1828:311-321) and in Oeuvres complètes 12 (1827:549-566), and then Jacobi (1849) and Jacobi's (1850) correction of Carlini, rather than the obscurities of Scheibner (who seems no clearer in his English paper than in his German). Laplace seems clearer in his expressed motivations for his work identifying the 'Laplace limit' of eccentricity, and his explanations of his method of identifying it, much clearer than the content of the Scheibner paper, and even the other papers. The limitations in the Laplace accounts are (a) his route to his results was mathematically non-rigorous as he admitted, and (b) he made a marginal error, arguably unimportant, in solving his equations numerically, obtaining not quite the right numerical value of the limit. Laplace's results themselves were supported by Jacobi in his own paper (1849) and in his correction and revision of Carlini (1850).

[3] The question also asks for derivations of the coefficients of these much-debated series expressed in terms of the mean anomaly. These can be found in Lagrange (1771), 'Sur le Problème de Képler'. Laplace gave at least some of them too, and there is also related briefer modern material in Brouwer and Clemence (1961) 'Methods of Celestial Mechanics' : 60-77.

There might be a further question, whether the Laplace limit has any astronomical significance? I would argue for an answer 'no'. Other series expansions, for the same quantities, are not subject to the limit. Examples include Euler's original 1747/9 series for the equation of the center, in terms of the eccentric anomaly: it clearly converges for all values of $e$ less than 1 'Recherches sur le mouvement des corps célestes en général': paper E112, at 116. (A similar series, this time for the true anomaly in terms of the eccentric anomaly, is given in Brouwer & Clemence 1961, p.63, eq.19.) Such examples make the limit itself look like a mathematical accident connected with a particular form of series expansion, rather than a reflection of any astronomical reality. Perhaps a knottier problem may be presented by the eccentric anomaly and the series representing solutions to Kepler's equation. Even if none of the known series for Kepler's equation are validly convergent for eccentricities near to 1, still there are well-established iterative methods for Kepler's equation, for E in terms of M, (e.g. Nijenhuis, 1991 as well as operationally simpler methods such as bisection and linear interpolation) undoubtedly good for high eccentricities also. In principle, these methods yield convergent sequences of approximations that can be differenced, providing terms that make a summable series convergent even at high elliptical eccentricities. Such series may be very ungainly and unsuited to explicit practical use, but they can still show the principle.