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

16-57] LOMMEL'S FUNCTIONS 547
and then the substitutions t = ceu and cosh u = r shew that
0f I b\ dt cos (x cosh u)du
cos at + = I
o o1 + t2 ceu + l/(ceu)
f00 f 1 1
cos (x cosh it) + du
cos (ceu + 1/(ce) +ce- + 1/(ce-)
a + b 0 cos (x.'). rdr
- Jx ( T2 + 72)(72 -)
Now consider
I f eix rTd
2ri (02+ T2) 2(7- )'
where F is a contour consisting of the real axis and a large semicircle above
it, the real axis being indented at T = + 1.
The only pole of the integrand inside the contour is at iO, and so
1 f eixT 7 d' e-x0
27 i r (02 + 2) V(r2- 1) 2i (02 + )'
As the radius of the large semicircle tends to infinity, the integral round
it tends to zero by Jordan's lemma, and hence
f0 cos (x). Td 1 f1 sin (xr). )drl 7re-x
(02 + 72) V(T2 1 (02 + 72) V(- 72) 2 24( + 1)
Thus we have
0cor ( b\ dt vre- a-b) a + b -fr sin (x cos q). cos do
cos at+ +.0ost t 1 +t2= 2 2x Jo 02+cOs2
But Cos 4c
But ^+eos2 == I+TC COS ~ C3 COS 3c + c5 cos 50 -...},
02+cosq2 1 + cc
and so we find that
(3) cos at + -)t2= - c2 m- J2m-l (+).
o t/ I r t2 n2=l
Similarly it is found that*
1 t 77-r e-(a-b) 00
(4) j sin (at + tdt - S c2J2 n(X),
I + t2 2 1
and
(5) Pj~iO COS b\ dt Isn. 1n
(5) Pf cos at+t)I_2= 7rsin(a + b)-7r ( —)m-l2-l 2m- (),
* Th detail otenyswle'm=1
(6) P sin at+) 1 t0 7- cos((a+ b)-7 - ()m-202mJ (a).
o deatlj 1 - thanlsswlbe foundinHrd=1l
* The details of the analysis will be found in Hardy's paper.