Asymptotic Approximation of Integrals (Classics in Applied by R. Wong

By R. Wong

I'm learning "Asyptotic Expantions" in graduate tuition. to check this box, this booklet is particularly very important reference considered one of them.

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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.

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