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

7-22]

ASYMPTOTIC EXPANSIONS

201

7'22. Stokes' phenomenon.
The formula ~ 721 (1) for J (z) was established for values of z such that
argz < t7. If we took arg z to lie between 0 and 27r (so that arg ze-" lies
between- 7r and 7r) we should consequently have
J, (z) = ev"i J4 (ze-i)
2 2 m. 1. ^ (-Y * (P 2,m)
ce ( Lc2 cos (ze~- 7i - V ) 2 (-)-.(' 2) r
\7-zej L 4 no (2ze-ri)2
* /sin W?((m. (P, 2m+ +1)]
_sm(^ -4.r-~4 ~o = 0 (2ze-R i)2
m=0 A
so that, when 0 < arg z < 27r,
Jv (z) co e(+') ri cos ( + 1 <J+ 1 /) (2 ()2 )
\ iz L 2 4 m:0 (2Z)21n
-sin (-) j (7, 2mn + 1)]
sin (z + lv7r + 77r) +,
21 =0 0 1)16 J
and this expansion is superficially quite different from the expansion of
~ 7'21 (1). We shall now make a close examination of this change.
The expansions of ~ 7'21 are derived from the formula
J, (z) = 2 {I[H( (z) + H z(2) (Z)},
and throughout the sector in which - 7r < arg z < 27r, the function HV() (z) has
the asymptotic expansion
'i( -f- 2, oo () 7r ( v m)2.
(2) W ~ =o (2z,^
The corresponding expansion for H?(2) (z), namely
J(2 )i (i)v,M
(1) H2 () C (-) e — (Z-2
() Z m=0(2iz)1?f'
is, however, valid for the sector - 27r< argz < t. To obtain an expansion
valid for the sector 0 < arg z < 2r we use the formula of ~ 3-62 (6), namely
Hf(2) (z) = 2 cos Vr. H(2 (ze-ri) + ev'i HfV) (ze-i),
and this gives
(2 ) H (Z) -)U ( e-iZ-7r-7r)
+ 2 cos v7r.( —) e ) (-M (, )
m=O (2z)'
The expansions (1) and (2) are both valid when 0<arg < 7r; now the
difference between them has the asymptotic expansion
2 cos Vr,. (- e:i(z++~2) mo (2) )
7=o (2/z)'
and, on account of the factor eiz which multiplies the series, this expression is
of lower order of magnitude (when z I is large) than the error due to stopping