A treatise on the theory of Bessel functions, by G. N. Watson.

17-2] KAPTEYN SERIES 553
In particular e sin E is an odd periodic function of M, and so, for all real
values of E, it is expansible into the Fourier sine-series
00
sin E= 2 A, sin rnM,
2 f- 1
where Ad = -- e sin E sin nMdM
7r /
[ 2E sin Ecos nMj7r 2 fw osd (Esin) EdM
== - ------- 4+ — cosrnM dM
[ nU Jo~nr 0?z' 0 dM
2 fw dE -dM dl M
= - cos nM dMi
n7r Jo d
2 T r
- cos nM. dE
nT7T o
2
=- J (ne).
n
Hence it follows that
" 2
(5) E = M+ E Jn (ne)sinnM,
n=l nl
and this result gives the complete analytical solution of Kepler's problem concerning the eccentric anomaly. The series on the right is a Kapteyn series which
converges rapidly when e < 1, and it is still convergent when e = 1; cf. ~~ 8'4, 8'42.
The radius vector is similarly expansible as a cosine series, thus
r ao
-= = B, + cos niM,
a n=l
where B, =1 (1-ecosE)dM
=1 F (1-ecosE)2dE
o
-1 1+ 2eoE
2 r2
while, when n = 0,
Bn =- (1-e cos E)cos nMdM
7 JO
= (-ecosE)sinnM + sin nM d (ecos E) dM
m'r o nro dM
2C= sin E sin (nE - ne sin E) dE
M7r o
2e
- rJn' (ne),
so that
(6) r 1 + 2- 2 2 J, (ne) cos nM.
ac w== ^