Elliptic expansions
The well-known Bessel functions of the first kind,
, where
is an integer, are defined as the
Fourier coefficients in the expansion of
:
 |
(A.116) |
It follows that
 |
(A.117) |
(Gradshteyn and Ryzhik 1980a).
The Taylor expansion of
about
is
 |
(A.118) |
for
(Gradshteyn and Ryzhik 1980b). Moreover,
In particular,
Let us write
 |
(A.125) |
where
is the eccentric anomaly, and
the mean anomaly, of
a Keplerian elliptic orbit. (See Section 4.11.)
It follows that
 |
(A.126) |
Integrating by parts, we obtain
 |
(A.127) |
However, according to Equation (4.59), the relationship between the eccentric and the mean anomalies is
 |
(A.128) |
where
is the orbital eccentricity. Hence,
![$\displaystyle A_n = \frac{1}{2\pi\,n} \oint {\rm e}^{-{\rm i}\,[(n-1)\,E - n\,e\,\sin E]}\,dE.$](img4044.png) |
(A.129) |
Comparison with Equation (A.117) reveals that
 |
(A.130) |
For the special case
, L'Hopital's rule, together with Equations (A.119) and (A.122), yields
 |
(A.131) |
where
denotes a derivative.
The real part of Equation (A.125) gives
where use has been made of Equations (A.120), (A.130), and (A.131). Likewise, the
imaginary part of Equation (A.125) yields
![$\displaystyle \sin E = \sum_{n=1,\infty}\left[\frac{J_{n-1}(n\,e)+ J_{n+1}(n\,e)}{n}\right]\sin(n\,{\cal M}).$](img4051.png) |
(A.133) |
It follows from Equations (A.121)–(A.124) that
Hence, from Equation (A.128),
According to Equation (4.69),
 |
(A.137) |
where
is is the radial distance from the focus of the orbit and
is the orbital major radius. Thus,
Equations (4.39) and (4.67) imply that
 |
(A.139) |
where
is the true anomaly. Hence, it follows from Equations (A.128) and (A.137), and the
fact that
when
, that
 |
(A.140) |
From Equation (A.136),
Thus,
and