uniform for large parameter
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21—30 of 36 matching pages
21: 10.69 Uniform Asymptotic Expansions for Large Order
22: 10.24 Functions of Imaginary Order
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10.24.1
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10.24.3
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►In consequence of (10.24.6), when is large
and comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv).
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►For mathematical properties and applications of and , including zeros and uniform asymptotic expansions for large
, see Dunster (1990a).
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23: 30.9 Asymptotic Approximations and Expansions
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§30.9(i) Prolate Spheroidal Wave Functions
… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). … ►The cases of large , and of large and large , are studied in Abramowitz (1949). …24: 10.21 Zeros
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§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros
… ►§10.21(vii) Asymptotic Expansions for Large Order
… ►§10.21(viii) Uniform Asymptotic Approximations for Large Order
… ►For derivations and further information, including extensions to uniform asymptotic expansions, see Olver (1954, 1960). The latter reference includes numerical tables of the first few coefficients in the uniform asymptotic expansions. …25: 10.45 Functions of Imaginary Order
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10.45.1
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10.45.6
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►In consequence of (10.45.5)–(10.45.7), and comprise a numerically satisfactory pair of solutions of (10.45.1) when is large, and either and , or and , comprise a numerically satisfactory pair when is small, depending whether or .
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►For properties of and , including uniform asymptotic expansions for large
and zeros, see Dunster (1990a).
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26: 2.4 Contour Integrals
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►Then
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►If this integral converges uniformly at each limit for all sufficiently large
, then by the Riemann–Lebesgue lemma (§1.8(i))
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►in which is a large real or complex parameter, and are analytic functions of and continuous in and a second parameter
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►For a coalescing saddle point and a pole see Wong (1989, Chapter 7) and van der Waerden (1951); in this case the uniform approximants are complementary error functions.
For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions.
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27: 10.20 Uniform Asymptotic Expansions for Large Order
28: Bibliography L
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The solutions of the Mathieu equation with a complex variable and at least one parameter large.
Trans. Amer. Math. Soc. 36 (3), pp. 637–695.
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Uniform asymptotic expansions of symmetric elliptic integrals.
Constr. Approx. 17 (4), pp. 535–559.
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Asymptotics of the first Appell function with large parameters II.
Integral Transforms Spec. Funct. 24 (12), pp. 982–999.
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Asymptotics of the first Appell function with large parameters.
Integral Transforms Spec. Funct. 24 (9), pp. 715–733.
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Asymptotic expansions of the Whittaker functions for large order parameter.
Methods Appl. Anal. 6 (2), pp. 249–256.
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