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coalescing saddle points


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1: 9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
2: 36.12 Uniform Approximation of Integrals
§36.12 Uniform Approximation of Integrals
§36.12(i) General Theory for Cuspoids
For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).
3: 12.16 Mathematical Applications
PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). …
4: 7.20 Mathematical Applications
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). …
5: 2.4 Contour Integrals
§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method
For two coalescing saddle points and an endpoint see Leubner and Ritsch (1986). For two coalescing saddle points and an algebraic singularity see Temme (1986), Jin and Wong (1998). For a coalescing saddle point, a pole, and a branch point see Ciarkowski (1989). For many coalescing saddle points see §36.12. …
6: Bibliography L
  • C. Leubner and H. Ritsch (1986) A note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points. J. Phys. A 19 (3), pp. 329–335.