By R. Wong

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**Extra info for Asymptotic Approximation of Integrals (Classics in Applied Mathematics)**

**Sample text**

Let where Jm(t) is the Bessel function of the first kind of order m. Use the identity (tmJm(t))' = tmJm-1(t) to show that This suggests that the integral I(x) may have the generalized asymptotic expansion 46 I Fundamental Concepts of Asymptotics Now, use the identity (Erdelyi et al. 1953b, p. 49) to show that as x -> thus proving the above generalized asymptotic expansion invalid. With the aid of the same identity, derive the expansion where Show that as -> , Rn = O(x-n-1/2). 9. , z) has the integral representation when Re a > 0 and Re z > 0.

The resulting expression is where Although this result is established when x is real and positive, by analytic continuation it also holds for complex x as long as Re x > 0. 32) with An = n + v + \ and z replaced by 2x yields 5. 31) and replacing x by z in the resulting expansion, we obtain as z -» oo in |arg z\ < n/2 — A. 33), we assume that v is real. 33) is bounded in absolute value by the first neglected term of the expansion. 32) through an angle y, where — Ti/2 < y < n/2. To do this, we note that for x > 0, we have by Cauchy's theorem The integral on the right-hand side actually defines an analytic function in — n/2 — y < arg x < n/2 — y.

9. , z) has the integral representation when Re a > 0 and Re z > 0. Derive the asymptotic expansion as |z| -> oo in |arg z| < n — d < n. Use the identity to remove the condition Re 10.