# distributional methods

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##### 1: 2.6 Distributional Methods

###### §2.6 Distributional Methods

… ►An important asset of the distribution method is that it gives explicit expressions for the remainder terms associated with the resulting asymptotic expansions. … ►The distribution method outlined here can be extended readily to functions $f(t)$ having an asymptotic expansion of the form …For a more detailed discussion of the derivation of asymptotic expansions of Stieltjes transforms by the distribution method, see McClure and Wong (1978) and Wong (1989, Chapter 6). … ►###### §2.6(iv) Regularization

…##### 2: Ingram Olkin

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►Olkin’s research covered a broad range of areas, including multivariate analysis, reliability theory, matrix theory, statistical models in the social and behavioral sciences, life distributions, and meta-analysis.
… Marshall), published by Academic Press in 1979,

*Statistical Methods for Meta-Analysis*(with L. … Hedges), published by Academic Press in 1985, and*Life Distributions: Non-Parametric, Semi-Parametric, and Parametric Families*(with A. …##### 3: 10.21 Zeros

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►For describing the distribution of complex zeros by methods based on the Liouville–Green (WKB) approximation for linear homogeneous second-order differential equations, see Segura (2013).
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##### 4: 3.5 Quadrature

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►In more advanced methods points are sampled from a probability distribution, so that they are concentrated in regions that make the largest contribution to the integral.
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##### 5: 3.8 Nonlinear Equations

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►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013).
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##### 6: Bibliography W

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The distribution of the zeros of Jacobian elliptic functions with respect to the parameter $k$
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Comput. Methods Funct. Theory 9 (2), pp. 579–591.
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##### 7: Bibliography O

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Tables of Fourier Transforms and Fourier Transforms of Distributions.
Springer-Verlag, Berlin.
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On the distribution of spacings between zeros of the zeta function.
Math. Comp. 48 (177), pp. 273–308.
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Asymptotic Enumeration Methods.
In Handbook of Combinatorics, Vol. 2, L. Lovász, R. L. Graham, and M. Grötschel (Eds.),
pp. 1063–1229.
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Computing $\pi (x)$: The combinatorial method.
Revista do DETUA 4 (6), pp. 759–768.
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Distribution of the partition function modulo $m$
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Ann. of Math. (2) 151 (1), pp. 293–307.
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##### 8: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals.
Numer. Algorithms 12 (3-4), pp. 259–272.
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Geometrical and Catastrophe Optics Methods in Scattering.
In Physical Acoustics, A. D. Pierce and R. N. Thurston (Eds.),
Vol. 21, pp. 1–234.
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Numerical Methods for Roots of Polynomials. Part I.
Studies in Computational Mathematics, Vol. 14, Elsevier, Amsterdam.
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The exponential integral distribution.
Statist. Probab. Lett. 5 (3), pp. 209–211.
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Methods and Programs for Mathematical Functions.
Ellis Horwood Ltd., Chichester.
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##### 9: Frank W. J. Olver

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►In 1945–1961 he was a founding member of the Mathematics Division and Head of the Numerical Methods Section at the National Physical Laboratory, Teddington, U.
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►Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history.
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►Having witnessed the birth of the computer age firsthand (as a colleague of Alan Turing at NPL, for example), Olver is also well known for his contributions to the development and analysis of numerical methods for computing special functions.
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##### 10: Bibliography K

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Numerical Methods and Software.
Prentice Hall, Englewood Cliffs, N.J..
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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The Rayleigh function: Theory and computational methods.
Zh. Vychisl. Mat. Mat. Fiz. 39 (12), pp. 1962–2006.
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An indirect method for evaluating certain infinite integrals.
Z. Angew. Math. Phys. 29 (3), pp. 380–386.
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Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution.
Justus von Liebig Verlag, Darmstadt (German, English).
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