relation%20to%20logarithmic%20integral

(0.005 seconds)

10 matching pages

1: 6.16 Mathematical Applications

… ►

§6.16(i) The Gibbs Phenomenon

… ►Hence, if

x

is fixed and

n\to\infty

, then

S_{n}(x)\to\frac{1}{4}\pi

0

, or

-\frac{1}{4}\pi

according as

0<x<\pi

x=0

, or

-\pi<x<0

; compare (6.2.14). … ►

§6.16(ii) Number-Theoretic Significance of $\operatorname{li}\left(x\right)$

►If we assume Riemann’s hypothesis that all nonreal zeros of

\zeta\left(s\right)

have real part of

\tfrac{1}{2}

(§25.10(i)), then …where

\pi(x)

is the number of primes less than or equal to

x

. …

2: 25.12 Polylogarithms

… ►The special case

z=1

is the Riemann zeta function:

\zeta\left(s\right)=\operatorname{Li}_{s}\left(1\right)

. ►

Integral Representation

… ►Further properties include …and … ►In terms of polylogarithms …

3: Bibliography M

… ►

A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

ⓘ

Notes:: Includes Fortran code, maximum accuracy 20S.
External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §6.13, 4th item, §6.21(ii).
Encodings:: BibTeX

… ►

I. Marquette and C. Quesne (2016) Connection between quantum systems involving the fourth Painlevé transcendent and

k

-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

ⓘ

External Links:: ISSN 0022-2488, Document, Link, MathReview Entry
Referenced by:: §18.38(iii).
Encodings:: BibTeX

… ►

R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

ⓘ

External Links:: FullText
Referenced by:: §8.24(i).
Encodings:: BibTeX

… ►

G. J. Miel (1981) Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.

ⓘ

External Links:: ISSN 0036-1429, Document, MathReview (M. Brannigan), ZentralBlatt
Referenced by:: §4.45(ii).
Encodings:: BibTeX

… ►

S. C. Milne (1985c) A new symmetry related to

\mathit{SU}(n)

for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

…

4: Bibliography

… ►

F. S. Acton (1974) Recurrence relations for the Fresnel integral

\int_{0}^{\infty}\frac{\exp(-ct)\,\mathrm{d}t}{\sqrt{t}(1+t^{2})}

and similar integrals. Comm. ACM 17 (8), pp. 480–481.

ⓘ

External Links:: ISSN 0001-0782, Document, MathReview, ZentralBlatt
Referenced by:: §6.18(ii).
Encodings:: BibTeX

… ►

S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

ⓘ

Notes:: Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.
External Links:: Code, Document
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

… ►

T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

ⓘ

External Links:: ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt
Referenced by:: (25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.
Encodings:: BibTeX

… ►

F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

ⓘ

External Links:: Document, MathReview (I. Marx), ZentralBlatt
Referenced by:: §29.8.
Encodings:: BibTeX

…

5: 20.11 Generalizations and Analogs

… ►As in §20.11(ii), the modulus

k

of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in

q

-series via (20.9.1). However, in this case

q

is no longer regarded as an independent complex variable within the unit circle, because

k

is related to the variable

\tau=\tau(k)

of the theta functions via (20.9.2). … ►For applications to rapidly convergent expansions for

\pi

see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). … ►For specialization to the one-dimensional theta functions treated in the present chapter, see Rauch and Lebowitz (1973) and §21.7(iii). … ►Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …

6: 25.6 Integer Arguments

… ►

§25.6(i) Function Values

… ►

25.6.11

\zeta'\left(0\right)=-\tfrac{1}{2}\ln\left(2\pi\right).

ⓘ

Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter and $\ln\NVar{z}$ : principal branch of logarithm function
Source:: Apostol (1985a, p. 223)
A&S Ref:: 23.2.13
Permalink:: http://dlmf.nist.gov/25.6.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §25.6(ii), §25.6 and Ch.25

►

25.6.12

\zeta''\left(0\right)=-\tfrac{1}{2}(\ln\left(2\pi\right))^{2}+\tfrac{1}{2}{% \gamma}^{2}-\tfrac{1}{24}\pi^{2}+\gamma_{1},

ⓘ

Symbols:: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\gamma_{\NVar{n}}$ : Stieltjes constants, $\pi$ : the ratio of the circumference of a circle to its diameter and $\ln\NVar{z}$ : principal branch of logarithm function
Source:: Apostol (1985a, pp. 226, 227, 229)
Referenced by:: Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series
Permalink:: http://dlmf.nist.gov/25.6.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §25.6(ii), §25.6 and Ch.25

… ►

25.6.15

\zeta'\left(2n\right)=\frac{(-1)^{n+1}(2\pi)^{2n}}{2(2n)!}\left(2n\zeta'\left(% 1-2n\right)-(\psi\left(2n\right)-\ln\left(2\pi\right))B_{2n}\right).

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function and $n$ : nonnegative integer
Source:: Miller and Adamchik (1998, (2), p. 202)
Referenced by:: (25.11.21)
Permalink:: http://dlmf.nist.gov/25.6.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §25.6(ii), §25.6 and Ch.25

… ►For related results see Basu and Apostol (2000).

7: Bibliography W

… ►

X.-S. Wang and R. Wong (2012) Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.

ⓘ

External Links:: ISSN 0219-5305, Document, Link, MathReview (Li Zhong Peng), ZentralBlatt
Referenced by:: §2.9(iii), §2.9(iii).
Encodings:: BibTeX

… ►

R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

… ►

B. M. Watrasiewicz (1967) Some useful integrals of

\mathrm{Si}(x),

\mathrm{Ci}(x)

and related integrals. Optica Acta 14 (3), pp. 317–322.

ⓘ

External Links:: Document, MathReview (S. D. Bajpai)
Referenced by:: §6.14(iii), §6.7(iv).
Encodings:: BibTeX

… ►

J. A. Wilson (1978) Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

ⓘ

Referenced by:: §16.4(iii), §16.4(iii), §16.4(iii).
Encodings:: BibTeX

… ►

J. Wimp (1984) Computation with Recurrence Relations. Pitman, Boston, MA.

ⓘ

External Links:: ISBN 0-273-08508-5, MathReview (S. Conde), ZentralBlatt
Referenced by:: §13.29(iv), §16.25, §2.9(iii), §3.10(iii), §3.6(iii), §3.6(v), §3.6(vii).
Encodings:: BibTeX

…

8: 14.30 Spherical and Spheroidal Harmonics

… ►

P^{m}_{n}\left(x\right)

and

Q^{m}_{n}\left(x\right)

(

x>1

) are often referred to as the prolate spheroidal harmonics of the first and second kinds, respectively. … ►

14.30.3

Y_{{l},{m}}\left(\theta,\phi\right)=\frac{(-1)^{l+m}}{2^{l}l!}\left(\frac{(l-m% )!(2l+1)}{4\pi(l+m)!}\right)^{1/2}e^{im\phi}\left(\sin\theta\right)^{m}\*\left% (\frac{\mathrm{d}}{\mathrm{d}(\cos\theta)}\right)^{l+m}\left(\sin\theta\right)% ^{2l}.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $m$ : integer and $l$ : nonnegative integer
Referenced by:: §14.30(ii)
Permalink:: http://dlmf.nist.gov/14.30.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

… ►

14.30.8

\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\overline{Y_{{l_{1}},{m_{1}}}\left(\theta,% \phi\right)}Y_{{l_{2}},{m_{2}}}\left(\theta,\phi\right)\sin\theta\,\mathrm{d}% \theta\,\mathrm{d}\phi=\delta_{l_{1},l_{2}}\delta_{m_{1},m_{2}}.

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $m$ : integer and $l$ : nonnegative integer
Referenced by:: §14.30(ii)
Permalink:: http://dlmf.nist.gov/14.30.E8
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.19):: As a notational clarification, the first spherical harmonic in the integral over $\theta$ been explicity marked up as a complex conjugate.
See also:: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

►See also (34.3.22), and for further related integrals see Askey et al. (1986). … ►

14.30.8_5

{\mathrm{e}}^{t{\mathbf{a}}\cdot{\mathbf{x}}}=\sqrt{4\pi}\sum_{n=0}^{\infty}% \sum_{m=-n}^{n}\frac{t^{n}r^{n}\lambda^{m}Y_{{n},{m}}\left(\theta,\phi\right)}% {\sqrt{(2n+1)(n+m)!(n-m)!}},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $n$ : nonnegative integer and $m$ : integer
Source:: Courant and Hilbert (1953, §VII.7.3)
Permalink:: http://dlmf.nist.gov/14.30.E8_5
Encodings:: pMML, png, TeX
Addition (effective with 1.1.0):: This equation was added.
Suggested 2019-09-26 by Anupam Garg
See also:: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

…

9: Bibliography C

… ►

F. Calogero (1978) Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial

{L}_{n}^{\alpha}(x)

as the index

\alpha\rightarrow\infty

and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

ⓘ

External Links:: ISSN 0024-1318, Document, MathReview (R. A. Askey)
Referenced by:: §18.7(iii).
Encodings:: BibTeX

… ►

B. C. Carlson (2006b) Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric

R

-functions. Math. Comp. 75 (255), pp. 1309–1318.

ⓘ

External Links:: ISSN 0025-5718, Document, MathReview, ZentralBlatt
Referenced by:: §19.25(v), §19.29(i), §22.14(ii).
Encodings:: BibTeX

… ►

H. S. Carslaw (1930) Introduction to the Theory of Fourier’s Series and Integrals. 3rd edition, Macmillan, London.

ⓘ

Notes:: Reprinted by Dover, New York, 1950.
External Links:: ZentralBlatt
Referenced by:: §6.16(i).
Encodings:: BibTeX

… ►

W. J. Cody (1991) Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

ⓘ

External Links:: ISSN 0098-3500, Document, MathReview, ZentralBlatt
Referenced by:: §5.24(ii), §5.24(iii).
Encodings:: BibTeX

… ►

S. W. Cunningham (1969) Algorithm AS 24: From normal integral to deviate. Appl. Statist. 18 (3), pp. 290–293.

ⓘ

Notes:: Includes Fortran program, maximum accuracy 12D.
External Links:: FullText
Referenced by:: §7.25(ii).
Encodings:: BibTeX

…

10: Software Index

… ►

Research Software.

This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

►

Open Source Collections and Systems.

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

►

Software Associated with Books.

An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

►

Commercial Software.

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►

Guide to Available Mathematical Software

A cross index of mathematical software in use at NIST.