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1: 36.4 Bifurcation Sets

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§36.4(i) Formulas

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Critical Points for Cuspoids

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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: 36 Integrals with Coalescing Saddles

Chapter 36 Integrals with Coalescing Saddles

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3: 36.12 Uniform Approximation of Integrals

§36.12 Uniform Approximation of Integrals

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§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).

4: 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. …

5: 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). …

6: 2.4 Contour Integrals

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§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method

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§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. …

7: Bibliography Q

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Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.

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External Links:

ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§2.4(vi).

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BibTeX

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8: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

Encodings:

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A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

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Unfolding the high orders of asymptotic expansions with coalescing saddles: Singularity theory, crossover and duality. Proc. Roy. Soc. London Ser. A 443, pp. 107–126.

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External Links:

ISSN 0962-8444, FullText, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§36.12(i), §36.12(ii), §36.12(iii).

Encodings:

BibTeX

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

Encodings:

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9: 20 Theta Functions

Chapter 20 Theta Functions

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10: 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). …