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

316

THEORY OF BESSEL FUNCTIONS

[CHAP. X

10'15. Asymptotic expansions of Anger- Weber functions of large order and
argument.
We shall now obtain asymptotic expansions, of a type similar to the
expansions investigated in Chapter vii, which represent J (z) and E, (z)
when I v and I z are both large.
In view of the results obtained in ~ 1013, it will be adequate to obtain
asymptotic expansions of the two integrals
_ eFvt-zsinhtdt.
r Jo
As in Chapter vIIi, we write
v = z cosh (a + i3) = z cosh 7,
where 0 ~ <3 < r and y is not nearly equal* to 7ri.
(I) We first consider the integral
1 I et-zsinh tdt = i J e-Z (t cosh 7+ sinh t) dt
r o 7r Jo
in which it is supposed temporarily that vz is-positive. When cosh y is positive,
t cosh y + sinh t steadily increases from 0 to oo as t increases from 0 to oo; we
shall take this function of t as a new variable r.
It is easy to shew that t is a monogenic function of r, except possibly when
r = (2n + 1) ri cosh y + sinh y ' y cosh y,
where n is an integer; and, when coshy is positive, none of these values of r is a real
positive number; for, when y is real, (2n + 1) rri cosh y does not vanish, and, when y is a pure
imaginary (=i/3), the singularities are on the imaginary axis and the origin is not one of
them since y is not equal to rri.
The expansion of dt/dr in ascending powers of T is
dt 0~
d7 m =o,T to
where a(O1 f(~+) 1 dt 1 f(+~) dt
~~where a~ 2-iJ TS2)T2d+1 dT d 2=i T2J +1
and so a, is the coefficient of l/t in the expansion of T-2~'-1 in ascending
powers of t. In particular we have
1 1 9 - cosh y
aO 1 + cosh al h2 (1 + cosh )4 a224(1 + cosh y)7
225 - 54 cosh y + cosh2 7y
a3 = 720 (1 + cosh y)10
From the general theorem of ~ 8'3, we are now in a position to write down
the expansion
1 (2m)! a,
(1~) fe__ e-vt-zsinhtdt 1 w)!a-
*EOasosvl' = / 0 2m+l1
* Expansions valid near -y=wri are obtained at the end of this section.