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

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BESSEL FUNCTIONS

51

A formula which is a kind of converse of (4), namely*
(5) p -( I' -(.-I)+ 1 ) j Y p _
~(5) ~ ~ (P- pif+z) Cr (V + 1) ()2 + 2) J ) zv,
in which P~J- denotes a generalised Legendre function, is due to Filon, Phil. Mag. (6) vi.
(1903), p. 198; the proof of this formula is left to the reader.
3'33. Gegenbauer's double integral of Poisson's type.
It has been shewn by Gegenbauert that, when R (v) > 0,
(lr)y W r
(1) Jv (w) = ( j f exp [iZ cos 0 - iz (cos 4 cos 0 + sin Q) sin 0 cos ~)]
sin2v-l sin2v Odq dO,
where 2 = Z2 + z2 - 2 Zz cos 4 and Z, z, 4 are unrestricted (complex) variables.
This result was originally obtained by Gegenbauer by applying elaborate integral transformations to certain addition formulae which will be discussed in
Chapter xi. It is possible, however, to obtain the formula in a quite natural
manner by means of transformations of a type used in the geometry of the
sphere.
After noticing that, when z = 0, the formula reduces to a result which is
an obvious consequence of Poisson's integral, namely
J, (Z)= (2)v eiz cos sin2v0. sini2v-1 d,
IF (V) J
we proceed to regard s and 0 as longitude and colatitude of a point on a
unit sphere; we denote the direction-cosines of the vector from the centre to
this point by (I, m, n) and the element of surface at the point by dw.
We then transform Poisson's integral by making a cyclical interchange of
the coordinate axes in the following manner~:
J (a)- ()-' C) eicose sinv 0 sin2 -1i #d0d#
7r7 (.)JoJo
-I )' ei(n e m2v-1 dco
7rr (V)JJ, 1
___(1_ff __ _ __
eial?i2V-1 is
- rr ()J) 1 o
= (2 )i o eiwsi 0 cos Cos2"-1 0 sin Odf dO.
' It is supposed that
aIzv r (v+1)zv-P'
az r(v-IU +l'
t Wiener Sitzuingsberichte, LxxIv. (2), (1877), pp. 128-129.
+ This method is effective in proving numerous formulae of which analytical proofs were
given by Gegenbauer; and it seems not unlikely that he discovered these formulae by the method
in question; cf. ~~ 12'12, 12'14. The device is used by Beltrami, Lombardo Rendiconti, (2) xm.
(1880), p. 328, for a rather different purpose.
~ The symbol Jfm>O means that the integration extends over the surface of the hemisphere on
which?n is positive.
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