# coalescing

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## 1—10 of 27 matching pages

##### 1: 36 Integrals with Coalescing Saddles

###### Chapter 36 Integrals with Coalescing Saddles

…##### 2: 9.15 Mathematical Applications

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►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.
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##### 3: 12.16 Mathematical Applications

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►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).
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##### 4: 31.13 Asymptotic Approximations

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►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999).
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##### 5: Christopher J. Howls

##### 6: 36.12 Uniform Approximation of Integrals

###### §36.12 Uniform Approximation of Integrals

►###### §36.12(i) General Theory for Cuspoids

… ►As $\mathbf{y}$ 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).##### 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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##### 8: Michael V. Berry

##### 9: 13.27 Mathematical Applications

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►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).
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##### 10: 2.4 Contour Integrals

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