asymptotic expansions of integrals
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31—40 of 93 matching pages
31: Bibliography S
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A simple approach to asymptotic expansions for Fourier integrals of singular functions.
Appl. Math. Comput. 216 (11), pp. 3378–3385.
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Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels.
SIAM J. Math. Anal. 11 (5), pp. 828–841.
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32: 2.5 Mellin Transform Methods
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►We are now ready to derive the asymptotic expansion of the integral
in (2.5.3) as .
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►The Mellin transform method can also be extended to derive asymptotic expansions of multidimensional integrals having algebraic or logarithmic singularities, or both; see Wong (1989, Chapter 3), Paris and Kaminski (2001, Chapter 7), and McClure and Wong (1987).
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►For examples in which the integral defining the Mellin transform does not exist for any value of , see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).
33: 19.20 Special Cases
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19.20.11
; () real.
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34: Bibliography H
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Asymptotic expansion of a class of integral transforms with algebraically dominated kernels.
J. Math. Anal. Appl. 35 (2), pp. 405–433.
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35: Bibliography D
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Error analysis in a uniform asymptotic expansion for the generalised exponential integral.
J. Comput. Appl. Math. 80 (1), pp. 127–161.
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36: 4.13 Lambert -Function
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4.13.10
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4.13.11
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►For these results and other asymptotic expansions see Corless et al. (1997).
►For integrals of use the substitution , and .
…For these and other integral representations of the Lambert -function see Kheyfits (2004), Kalugin et al. (2012) and Mező (2020).
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37: 29.16 Asymptotic Expansions
§29.16 Asymptotic Expansions
►Hargrave and Sleeman (1977) give asymptotic approximations for Lamé polynomials and their eigenvalues, including error bounds. The approximations for Lamé polynomials hold uniformly on the rectangle , , when and assume large real values. …38: 9.13 Generalized Airy Functions
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►As
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§9.13(ii) Generalizations from Integral Representations
… ►and the difference equation … ►These properties include Wronskians, asymptotic expansions, and information on zeros. … ►39: 2.2 Transcendental Equations
§2.2 Transcendental Equations
… ►An important case is the reversion of asymptotic expansions for zeros of special functions. … ►
2.2.7
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►where and () is the coefficient of in the asymptotic expansion of (Lagrange’s formula for the reversion of
series).
…For other examples see de Bruijn (1961, Chapter 2).
40: 5.19 Mathematical Applications
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►By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of for large , or small , can be obtained complete with an integral representation of the error term.
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