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31: 14.15 Uniform Asymptotic Approximations
§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
… ►32: 1.10 Functions of a Complex Variable
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►The convergence of the infinite product is uniform if the sequence of partial products converges uniformly.
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-test
… ►where is analytic for all , and the convergence of the product is uniform in any compact subset of . …33: 12.14 The Function
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§12.14(ix) Uniform Asymptotic Expansions for Large Parameter
… ►Airy-type uniform asymptotic expansions can be used to include either one of the turning points . … ►Positive ,
… ►Airy-type Uniform Expansions
… ► …34: Bibliography L
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A note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points.
J. Phys. A 19 (3), pp. 329–335.
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A uniform asymptotic expansion for Krawtchouk polynomials.
J. Approx. Theory 106 (1), pp. 155–184.
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Generalized Riemann -function regularization and Casimir energy for a piecewise uniform string.
Phys. Rev. D 44 (2), pp. 560–562.
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Uniform asymptotic expansions of symmetric elliptic integrals.
Constr. Approx. 17 (4), pp. 535–559.
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Uniform asymptotic expansions at a caustic.
Comm. Pure Appl. Math. 19, pp. 215–250.
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35: Bibliography P
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On two-sided estimates, uniform with respect to the real argument and index, for modified Bessel functions.
Mat. Zametki 65 (5), pp. 681–692 (Russian).
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Error bounds for the uniform asymptotic expansion of the incomplete gamma function.
J. Comput. Appl. Math. 147 (1), pp. 215–231.
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A uniform asymptotic expansion for the incomplete gamma function.
J. Comput. Appl. Math. 148 (2), pp. 323–339.
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36: 36.13 Kelvin’s Ship-Wave Pattern
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►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 .
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37: Bibliography B
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Application of uniform asymptotics to the second Painlevé transcendent.
Arch. Rational Mech. Anal. 143 (3), pp. 241–271.
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Uniform approximation for potential scattering involving a rainbow.
Proc. Phys. Soc. 89 (3), pp. 479–490.
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Uniform approximation: A new concept in wave theory.
Science Progress (Oxford) 57, pp. 43–64.
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Uniform asymptotic smoothing of Stokes’s discontinuities.
Proc. Roy. Soc. London Ser. A 422, pp. 7–21.
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Uniform asymptotic expansion of Charlier polynomials.
Methods Appl. Anal. 1 (3), pp. 294–313.
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38: 2.8 Differential Equations with a Parameter
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►The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to .
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►The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to .
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►The expansions (2.8.25) and (2.8.26) are both uniform and differentiable with respect to .
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►The expansions (2.8.29) and (2.8.30) are both uniform and differentiable with respect to .
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►For two coalescing turning points see Olver (1975a, 1976) and Dunster (1996a); in this case the uniform approximants are parabolic cylinder functions.
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39: 7.20 Mathematical Applications
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►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).
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