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

548

THEORY OF BESSEL FUNCTIONS

[CHAP. XVI

The last two results may be written in the form
(7)^~~ _T/ T 2 _ 1' x2t w\ _t
(7) 1U((w,x)+1,(, )=- P os (CS -t 1 dt+
7T 0 2 2t/ - t2
(8) TT / X -^ / ^. 2 0 2 t W\ tdt
(8) U (w, x) + VO (, x)= -r si ( + t
provided that 0 < w < x.
16'58. Integrals of Gilbert's type for Lommel's functions.
An obvious method of representing U (w, z) and V (w, z) by integrals is
to substitute the Bessel-Schlafli integral of ~ 6'2 for each Bessel function in
the appropriate series. We thus get
r ~~ r->~< 1..,,\~t-ami (o+) / 2 2 dt
u,(w )v Z (W-)M (1-w)0+2m exp (t- dt
rr2 i 0=O 2 -( oo ____
When the contour is so chosen that it lies wholly outside the circle on
which t = l1 w, we may change the order of summation and integration
and get
(1) U, (w, ) = 2i ( w+, it+) (j W/et) ( - 2 dt
+"/e
Now the residues of the integrand at + Iiw are
iw iZ v_ rri
2exp l+~2+~2+ -2 '
and so
V ( ) 1 [(0+) (I w/ov 22 dt
Y 2 (W,.) Z j + IW2/t2 aexp( t 4-.t
Making a slight change in the notation, we deduce that
1 +) (tlw) /t Z\dt
~(2) Vv^^',.2i 1 +t/W2 (exp2 2t- t'
and, in this integral, the points + iw lie outside the contour.
In general it is impossible to modify the contour in '(2) into the negative
half of the real axis taken twice, in consequence of the essential singularity of
the integrand at the origin. The exception occurs when z= 0, because then
the essential singularity disappears, and
(3) V(wO)= 1i ( +)(t/w)V et dt
V~ ( ~,, 0) J _ tw + t//W t
and hence
Osin vr doo exp ia 2v-le-AuW
(4) V (w, O)= sin exp u- e1-atu du,
provided that R(v) > 0 and a is an acute angle (positive or negative) such
that
a + arg w | < 2 77.