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

16-54, 16-55] LOMMEL S FUNCTIONS 543
To appreciate the significance of such relations observe that
dt cos (2wt). uv ( zt) + sin (-t2). U ~+ (w zt)j
-sin ( wt2). [ -,Ju (-, zt) - w t.it2 (j - zt)
On integration we find that
(1) - J (zt) sin (Wwt2) t1- dt
=- cos 2w Uv - 7 z)-sin-w. o), ( ) ) + Uv ( ~)'
Hence it follows that
(3) Zz-;Zi J~_ (zt) cos { w (1 - t2)}. tv dt
= - 2 w ) + cos W.U2(
and
(4) -,W-l SJ~_J (zt) sinn { w (1 - t2) ). t dt
= U (_ (, ) - cos -w. U_^ (, 0 + sin w. U2_ (i, O )
and these integrals differ from the corresponding integrals of the preceding
section only in te sign of the order of the Bessel function.
The reader will find some additional formulae concerning Lommel's functions in a
paper by Schafheitlin, Berliner Sitzungsberichte, VIII. (1909), pp. 62-67.
1655. Psedo-addtion forlae for fnctios of orders and
Some very curious formulae have been obtained by Lomel, which connect
functions of the type Uv (w, z) with functions of the same type in which the
second variable is-zero, provided that v is equal to W or
and, similarly,
When we write v = - in ~16'53 (5), we get
2 2
Hence it follows that
U ((, (z) ~ )2 i(exp )+ = (( f + t - - wt2)} dt.
sectio Jon (zt)i sin oI the (I of t ihe f tio t d dt
paper COS ScaW. n B e, 0S + Sin vW ( p2-p. 6-7
16-55. Pseudo-addition formulae for functions of orders I and 3.
Some very curious formulae have been obtained by Lommel, which connect
functions of the type U, (w, z) with functions of the same type in which the
second variable is zero, provided that v is equal to I or 3.
When we write v = ^ in ~ 16-53 (5), we get
UY, (W z) ~ (w,) == C-exp J~ i {^w - zt - I Wt2)} dt
) \i Ua 22 2 -
+,- exp{~ i('W zt Wtdt.
2Z7^/ )