uniform asymptotic approximation
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§14.26 Uniform Asymptotic Expansions►The uniform asymptotic approximations given in §14.15 for and for are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). … ►See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.
§28.26 Asymptotic Approximations for Large… ►
§28.26(ii) Uniform Approximations…
… ►Close to the bifurcation set but far from , the uniform asymptotic approximations of §36.12 can be used. …
§28.8 Asymptotic Expansions for Large… ►
Barrett’s Expansions… ►
Dunster’s Approximations►Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). … ►
§14.15 Uniform Asymptotic Approximations►
§14.15(i) Large , Fixed… ►For asymptotic expansions and explicit error bounds, see Dunster (2003b). ►
§14.15(iii) Large , Fixed… ►
… ►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … ►If has a double zero , or more generally is a zero of order , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order . …
… ►The disturbance can be approximated by the method of uniform asymptotic approximation for the case of two coalescing stationary points (36.12.11), using the fact that are real for and complex for . … ► …
… ►For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951). …
Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow.
Studies in Appl. Math. 53, pp. 91–110.
Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory.
Studies in Appl. Math. 53, pp. 217–224.
… ►This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. …