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

350 THEORY OF BESSEL FUNCTIONS [CHAP. X
10'74. Functions expressible in terms of Lommel's functions.
From the descending series given in ~ 10'71 (1) it is evident that Neumann's
polynomial On(z) is expressible in terms of Lommel's functions by the equations
(1) 02 (z) = (l/z) S,/S (z), 02,,+ (z) = {(2m + 1)/z} S0, ml (z),
and Schlafli's polynomial Sn (z) is similarly expressible by the equations
(2) Sm (z) = 4m S-_, 2m (Z), 2+, (z) = 2S0,2M1+l (z).
It is also possible to express the important integrals
-, | (z) dz,, rz (z) dz
in terms of Lommel's functions; thus we have
d -Jz J, (z). z-v S-l, ^- (z)} = ZJ-1 (z) S_ P-l (z) + (p - V - 1 ) J(S^ z-, (z),
dZ -^ JV-, (z). Z- S&,. (Z)} =- ZJ (z) S, v (z) + (/ + v - 1) ZJV_ () S_ v-l (z).
On eliminating S,_^ _-(z) from the right of these equations, and using
~ 10'72 (6), we find by integrating that
(3) zJ Jv (z) dz = (Ca + v - 1) zJv (z) Sv- (z) -zJ^_x (z) S, v (z),
and proofs of the same nature shew that
(4) f z- Y7 (z) dz = (/, + v - 1) zTY (z) S -,^_i (z) - YV, (z), (z),
and, more generally,
(5) J' W (z) dz = (t + I - 1) zV (z) S-1, -_ (z) - zV-, (z) S, (z).
Special cases of these formulae are obtained by choosing / and v, so that
the functions on the right reduce to Neumann's or Schlafli's polynomials, thus
(6) zW2 rs (z) daz = (x O2.-n (z ) 0 - 2_, - () - (z) 021 (z)
(7) 2z,,4i (z) dz = {2m+, ) 1 ( z) -, (z) - 2 m+ (z) 2, )}.
Of these results, (1), (3), (4) and (6) are contained in Lommel's paper, Math. Ann. Ix.
(1876), pp. 425-444; (6) and (7) were given by Nielsen, Handbuch der Theorie der
Cylinderfuniktionen (Leipzig, 1904), p. 100, but his formulae contain some misprints.
It should be noticed that Lommel's function, in those cases when it is