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

374

THEORY OF BESSEL FUNCTIONS

[CHAP. XII

grand in powers of z and integrating term-by-term, thus
zv+l ram
2 r (v + 1)J
2) r 1) f7i-j J JY~ (z sin 0) sinlh+i 0 Cos2v+~ dO
~~ (_)m,(-) +v+2M zm+ i
(.. 29 _+2 ) m! I+ M __ v + sin y+2+im1 0 cosiv+l OdO
=0 2-+~+2m! r( m+ l)r (.+l) 1)Jo
(oo (\m) (I2)++2m++l
=O0 m!r(/u+^+ +m+2)
and the truth of the formula is obvious.
It will be observed that the effect of the factor sin"+l 0 in the integrand
is to eliminate the factors F (/u + m + 1) in the denominators. If we had taken
sin'-e 0 as the factor, we should have removed the factors m!. Hence, when
R (v) >- 1 and / is unrestricted, we have
(2) JY (z sin 0) sinl- 0 cos+l OdO = S+-1, v_1, (z)
In particular, by taking v = -, we have
(3) (2zI / ir J i (z s sin)sin' Od = H_ (z).
A formula* which is easily obtained from (1) is
(4) fJ (z sin 0) v (z cos 0) tan +1 0d0 = r ( v^- I J (z)
JO~~~~~when RI2+2+ () > R (-1 + )
when 1? (v) > R (/,) > - 1. This may be proved by expanding 7 (z cos 0) and
integrating term-by-term, and finally making use of Lommel's expansion
given in ~ 5'21.
The functional equation, obtained from (1) by substituting functions to be determined,
F, and F + +l, in place of the Bessel functions, has been examined by Sonine, Math. Ann.
LIX. (1904), pp. 529-552.
Some special cases of the formulae of this section have been given by Beltrami,
Istituto Lombardo Rendiconti, (2) xnII. (1880), p. 331, and Rayleigh, Phil. Nfag. (5) xII. (1881),
p. 92. [Scientific Papers, I. (1899), p. 528.]
It will be obvious to the reader that Poisson's integral is the special case of (1) obtained
by taking / = -.
For some developments of the formulae of this section, the reader should consult two
papers by Rutgers, Niezuw Archiefvoor Wiskunde, (2) vi. (1905), pp. 368-373; (2) vii. (1907),
pp. 88-90.
12'12. The geometrical proof of Sonine's first integral.
An instructive proof of the formula of the preceding section depends on
the device (explained in ~ 3'33) of integrating over a portion of the surface of
a unit sphere with various axes of polar coordinates.
If (1, m, n) are the direction cosines of the line joining the centre of the

* Due to Rutgers, Nieuw Archief voor Wiskunde, (2) vII. (1907), p. 175.