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coalescing transition points

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1: 36.4 Bifurcation Sets
§36.4(i) Formulas
Critical Points for Cuspoids
Critical Points for Umbilics
This is the codimension-one surface in x space where critical points coalesce, satisfying (36.4.1) and … This is the codimension-one surface in x space where critical points coalesce, satisfying (36.4.2) and …
2: 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. …
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: 36.12 Uniform Approximation of Integrals
§36.12 Uniform Approximation of Integrals
§36.12(i) General Theory for Cuspoids
As y varies as many as K + 1 (real or complex) critical points of the smooth phase function f can coalesce in clusters of two or more. … In (36.12.10), both second derivatives vanish when critical points coalesce, but their ratio remains finite. … 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).
5: Bibliography Q
  • W.-Y. Qiu and R. Wong (2000) Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.
  • 6: 2.8 Differential Equations with a Parameter
    §2.8(ii) Case I: No Transition Points
    For connection formulas for Liouville–Green approximations across these transition points see Olver (1977b, a, 1978).
    §2.8(vi) Coalescing Transition Points
    For two coalescing turning points see Olver (1975a, 1976) and Dunster (1996a); in this case the uniform approximants are parabolic cylinder functions. … For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order. …
    7: 2.4 Contour Integrals
    §2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method
    §2.4(vi) Other Coalescing Critical Points
    For two coalescing saddle points and an endpoint see Leubner and Ritsch (1986). …For a coalescing saddle point, a pole, and a branch point see Ciarkowski (1989). For many coalescing saddle points see §36.12. …
    8: 13.27 Mathematical Applications
    For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i). …
    9: Bibliography O
  • A. B. Olde Daalhuis (2000) On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.
  • F. W. J. Olver (1964b) Error bounds for asymptotic expansions in turning-point problems. J. Soc. Indust. Appl. Math. 12 (1), pp. 200–214.
  • F. W. J. Olver (1975a) Second-order linear differential equations with two turning points. Philos. Trans. Roy. Soc. London Ser. A 278, pp. 137–174.
  • F. W. J. Olver (1976) Improved error bounds for second-order differential equations with two turning points. J. Res. Nat. Bur. Standards Sect. B 80B (4), pp. 437–440.
  • F. W. J. Olver (1977c) Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.
  • 10: 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). … Let P ( t ) = P ( x ( t ) , y ( t ) ) be any point on the projected spiral. …