ABSTRACT
The problem is to find the asymptotic form of F,(x) for fixed positive p and large positive x. Special cases (which are useful later as checks) include Fl(x) = e' and Fz (x) = 1, (2x), where I denotes the modified Bessel function. 8.2 Consider the contour V of Fig. 8.1, in which n denotes an arbitrary positive integer. By use of the residue theorem and equations (7.05) and (7.06)' we derive
where V1 and %', are the upper and lower halves of %', respectively. With the aid of Stirling's approximation (4.04) we may verify that the integrals
where the path for the second integral consists of the quarter circle DC and the imaginary axis from C to ioo. On this path-and also in the interval [-+,O]-we have IxtPJ f I when x 2 1. Therefore
The asymptotic behavior of the last integral is derivable by Laplace's method, as follows. Since
and +(t+ I) is an increasing function, the integrand has a single peak, located at the root of the equation
+(t+l) = Inx. For large x this root is given by t - x; compare Exercise 4.2. With a view to replacing T(t+ 1) by its Stirling approximant, we subdivide the integration range at Ax, where A is an arbitrary constant in the interval (0,l).
