
\documentstyle[12pt]{article}
\textheight 240mm
\textwidth  165mm

\begin{document}
\pagestyle{empty}
\baselineskip 16pt
\parindent 0pt
\parskip   5pt

\begin{center}
{\large \bf D'Alembertian  functions in Celestial Mechanics as
a space of holomorph--in--a--bottle functions }\\

\smallskip

{\bf K.V. Kholshevnikov}\\

{\sl Astronomical Institute, St. Petersburg University,\\
Stary Peterhof, 198904 St.~Petersburg, Russia.\\
E-mail: kvk@aispbu.spb.su}
\end{center}

\noindent
\rm
The well--known D'Alembertian characteristic of disturbing and other
important for Celestial Mechanics functions means a property of Fourier
series coefficients, permitting to replace it by a Maclaurin series in
terms of  $ x = r e^{i \phi} , y = r e^{-i \phi} ,\phi $ being an angular
variable, $ r $~--- action one (e.g. eccentricity or inclination).
{\em We show that such a coefficients property is a consequence of a
functional one}. Namely, introduce a notion of D'Alembertian functions
which are holomorph and limited in a domain $ 0 < |r| e^{| Im \phi|} < R $
and are invariant under the transformation
$ r , \phi\; \rightarrow\; -r , \phi + \pi $.
Such a domain may be represented like a bottle with infinitely
long and infinitely thin neck.

Of course a D'Alembertian function possesses a D'Alembertian
characteristic, moreover we can estimate the coefficients, the size
of convergence domain and the function itself.

{\em Simpliest example}. Consider Cartesian coordinates of a planet in
elliptic motion. Let semi--major axis is arbitrary, mean longitude
varies  in a $ 2c $ wide strip symmetric with respect to real axis. Then
the coordinates are D'Alembertian functions with respect to two pairs:
sine of semi--inclination ~--- longitude of ascending node; eccentricity~---
longitude of pericentre. In the first case $ R = 1 $, in the second
$ R = 1 / {ch}\, (c + 1) $ . Considering only real values we may put
$ c = 0 $, so $ R = 0.648 $ that is a little bit less than a famous
{\em Laplace limit } $ 0.663 $. To reach it we ought to introduce a
complicated asymmetrical bottle instead of a simple one above, that
appears to be unnecessary .

\end{document}
