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

8.4j FUNCTIONS OF LARGE ORDER 243
Now d (w, -w) r
dT m = 20 -ar I,=o
when i 7 is sufficiently small; and since
dT
- = cosh a - cosh w,
dw
it follows that d (wz - w2)/d tends to zero as T tends to + oc.
Hence the conditions stated in the lemma of ~ 8'3 are satisfied, and so
o e { dr dwr jd
has the asymptotic expansion
ct21n Fr (in + ),>E + 2
m 3 p=0 xm+~
when x is large.
Since arg {(wi - a)/rT -I v as T -- 0, it follows that, in (2), the phase of ao
has to be interpreted by the convention arg a0 = + 7r, and hence
ev(tanh a-a) F F(m~+ l) Ai
(3) Jr (v seeh a) /(2-rv tanh a) F =o (~) '(1v tanh a)m'
where
o0=1, A= —,= ~ —coth2a,
(4) A2 = T28- 57 coth2 a + 3-45 _ coth4,
The formula (3) gives the asymptotic expansion of J (y sech a) valid when
a is any fixed positive number and v is large and positive.
The corresponding expansion for the function of the second kind, obtained by taking a
contour from - o to oo + ri, is
ev(a- tanh a) r(m+~) (_)rA
(5) Yr (v sech a)- - a ) ('t./(Qtwv tanh a) m0P=o (i) ( v tanh a)""
The position of the singularities of d (w, -1w)/dr, qua function of the
complex variable 7, should be noted. These singularities correspond to the
points where w fails to be a monogenic function of T, i.e. the points where
dr/dw vanishes. Hence the singularities correspond to the values + a + 2nri
of w, so they are the points where
T = 2n7ri cosh a, z= 2 (sinh a - a cosh a) + 2ni7r cosh a,
and i assumes all integral values.
It is convenient to obtain a formula for dw/dr in the form of a contour integral; if
(wo, r0) be a pair of corresponding values of (w, r), then, by Cauchy's theorem,
(dw\ 1 ^(Tr) dv dr 1 (w) _
\dTro 2ri J dr r-To 27ri J T- T(
where the contour includes no point (except W0) at which r has the value ro.
1.6-2