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

7-4] ASYMPTOTIC EXPANSIONS 217
Changing the sign of / is equivalent to replacing u by 1/zt, and so, replacing
the expression on the left by its value as a function of the third kind, we have
~1 f ('- ~ ~ +1ia x /, 1J cd log it
(3) e Hvif(l (rei(a-) -= - (^ + u) exp +re ( +-) d
trz J o 2/u) do
From (2) it follows that - rei (u- )2/- increases steadilyn from 0 to + X
as ( varies monotonically from 0 to 7i - a; and, if we write
-re ( -_ 1)2/u =t,
so that t is positive when u is on the contour, we have
du dt dt
uL. -reio (u - 1/u) - e-s(-"a)i (md + ~6-) /(rt)'
the range of values of arg i being less than 7r.
Next, by Cauchy's theorem,
t1 + it 2i f { ~ } -;
27TZn + U- -1/uj
it is convenient to take the point = 1 inside the contour, but 0 = 0 must be
outside the contour because the origin is a branch-point.
It follows that
U= + U,-j (u+, 1/U+) 1v-i (-_ 1) dl_
(4) UV + u —"= 2
27i i ( - 1)2 + S't/(reia)'
Hence
(5) Hv() (r ef (a — r)). e-ri expr (re) (+l/u+, l+) e-t t-t rv- (-_ 1) drdt
(5) H f(1) (rei")- 2ri r(eOi7. ('- 1)2 + 't/(rei) )
Now it is evident that
1 p-1 (._.)m ~ -1n (_)P 'P tP
(- 1)2 + t/(rei) -,, (- 1)2m+"2 i (rei)) (P- )(e {(f- )+ tl/(rea)}'
where p is any positive integer (zero included). It will be convenient subsequently to suppose that p exceeds both R (v - ~) and R (- v - ).
On making this substitution in the last integrand and observing that
1(2+u+31++) 12F^r (v ~+ V + )) _!!(., m)
27T (+'riJ +'~+) r.+m-J ( - m = _. ( +) (2mn!) (2m)!
(with the notation of ~ 7'2), we deduce that
2 \ P1 (-4 (. mi M
(6) o> (e(ei -(a))= i7 ) exp (rei - 7ri) [. ()' - ( R)
\ ewvre1~ / m (2reia )n
where
Rp1) HP (-) f r (ut, +l/, 1+I) e-it tp-i v^+P- d'dt
P 27rV(27r)JJ (- 1)2P-1i (reta)P {(- 1)2 + 't/(reia)}
* Since d sin2 b sin (1 +2 cos a cos 0 + cos2 ')
~d cs +cos (cosacosce )2
d~6 cos a + cos <f> (cos a + cos 0)2