special distributions

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31: 25.16 Mathematical Applications

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§25.16(i) Distribution of Primes

►In studying the distribution of primes $p\le x$, Chebyshev (1851) introduced a function $\psi \left(x\right)$ (not to be confused with the digamma function used elsewhere in this chapter), given by … ► $H\left(s\right)$ is the special case $H(s,1)$ of the function … ►

25.16.14 $$\sum _{r=1}^{\mathrm{\infty }}\sum _{k=1}^r\frac{1}{rk(r+k)}=\frac{5}{4}\zeta \left(3\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function and $k$: nonnegative integer

Keywords:

infinite series, special value

Source:

Apostol and Vu (1984, (14), p. 94)

Permalink:

http://dlmf.nist.gov/25.16.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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25.16.15 $$\sum _{r=1}^{\mathrm{\infty }}\sum _{k=1}^r\frac{1}{r^2(r+k)}=\frac{3}{4}\zeta \left(3\right).$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function and $k$: nonnegative integer

Keywords:

infinite series, special value

Source:

Apostol and Vu (1984, (15), p. 94)

Permalink:

http://dlmf.nist.gov/25.16.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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32: 3.5 Quadrature

… ►A special case is the rule for Hilbert transforms (§1.14(v)): … ►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. …

33: Bibliography M

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Algorithm 757: MISCFUN, a software package to compute uncommon special functions. ACM Trans. Math. Software 22 (3), pp. 288–301.

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Notes:

Double-precision Fortran, maximum accuracy 20D.

External Links:

ISSN 0098-3500, Document, GAMS (module 757; package TOMS), MathReview Entry, ZentralBlatt

Referenced by:

5th item, 1st item, 1st item, 4th item, 1st item, 1st item.

Encodings:

BibTeX

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Asymptotic expansions for the zeros of certain special functions. J. Comput. Appl. Math. 145 (2), pp. 261–267.

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External Links:

ISSN 0377-0427, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.70, §11.4(vii), §6.13.

Encodings:

BibTeX

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A Handbook of Generalized Special Functions for Statistical and Physical Sciences. Oxford Science Publications, The Clarendon Press Oxford University Press, New York.

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External Links:

ISBN 0-19-853595-3, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§16.18, §16.19, §16.20.

Encodings:

BibTeX

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On the evaluation of indefinite integrals involving the special functions: Application of method. Quart. Appl. Math. 13, pp. 84–93.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§14.17(i).

Encodings:

BibTeX

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The exponential integral distribution. Statist. Probab. Lett. 5 (3), pp. 209–211.

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External Links:

ISSN 0167-7152, Document, MathReview Entry, ZentralBlatt

Referenced by:

§8.19(xi).

Encodings:

BibTeX

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34: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►In this section we will only consider the special case $w(x)=1$, so $d\alpha (x)=dx$; in which case $L^2\left(X\right)\equiv L^2(X,dx)$. … ►of the Dirac delta distribution. … ►The special form of (1.18.28) is especially useful for applications in physics, as the connection to non-relativistic quantum mechanics is immediate: $-\frac{d^2}{{dx}^2}$ being proportional to the kinetic energy operator for a single particle in one dimension, $q(x)$ being proportional to the potential energy, often written as $V(x)$, of that same particle, and which is simply a multiplicative operator. … ►For a formally self-adjoint second order differential operator $ℒ$, such as that of (1.18.28), the space $𝒟({ℒ}^{\ast })$ can be seen to consist of all $f\in L^2\left(X\right)$ such that the distribution $ℒf$ can be identified with a function in $L^2\left(X\right)$, which is the function ${ℒ}^{\ast }f$. …