ABSTRACT
The only minimum ofp(a, 8) in the range of integration is at 8 = a. By Laplace's method it is readily verified that for fixed a and large x the asymptotic expansion of I(a, x) begins
We apply the method of 59 to derive an asymptotic expansion for Z(a, x) which is uniformly valid for a E [O,ao], where a, is any fixed number in the interval 0 < a, < $TI. Since
we have, in the notation of $9.2, a = 2'I2(cosa+a sina-1)'12, (10.04)
cos8 + 8 sina = 1 + aw - +w2, (10.05) and
~ = a _ + 2 ~ ~ ~ { c o ~ a + ( a - 8 ) s i n a - c o s 8 } ' ~ ~ ( 8 2 ~ ) . The transformed integral is given by
where
and
dw sine-sina'
so that
compare (9.06). Leading coefficients are easily calculated by substituting in (10.07)
348 9 Integrals: Further Methods
and equating coefficients. The first three are found to be 1 sin a 5-2 cos2a
4o(a) = (cosa)1/2 9 41(a) = - 3 C O S ~ a (a) = 24 (cOs a)7/z . (10.1 1)
The next step is to replace the upper limit in (10.06) by. infinity, substitute for dO/dw by means of (10.10), and integrate formally term by term. Making the substitution w = u + ( ~ / x ) ' / ~ T and referring to (10.04), we see that
where
y denoting the incomplete Gamma function (Chapter 2,851). Therefore the desired expansion is given formally by
r (+s+t) x,(a, x) < 2r(4s+4) (s even); 0 < X,(a, x) c r(f~++) (s odd). This suggests that the series (10.14) may be a generalized asymptotic expansion with scale {e~(c~sa+-ina)x -(s+')12}. Before embarking on the proof we confirm that we are on the right track by examining the reduced form of (10.14) in cases when a is fixed.
