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21: 10.9 Integral Representations

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Poisson’s and Related Integrals

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Schläfli’s and Related Integrals

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Mehler–Sonine and Related Integrals

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§10.9(ii) Contour Integrals

… ►See Paris and Kaminski (2001, p. 116) for related results. …

22: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

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§19.2(ii) Legendre’s Integrals

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§19.2(iii) Bulirsch’s Integrals

… ►Lastly, corresponding to Legendre’s incomplete integral of the third kind we have … ►

§19.2(iv) A Related Function: $R_C(x,y)$

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23: 3.5 Quadrature

… ►For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. … ►The monic and orthonormal recursion relations of this section are both closely related to the Lanczos recursion relation in §3.2(vi). … ►are related to Bessel polynomials (§§10.49(ii) and 18.34). … … ►The standard Monte Carlo method samples points uniformly from the integration region to estimate the integral and its error. …

24: Bibliography S

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A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.

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External Links:

ISSN 0036-1410, Document, MathReview (Ernest G. Kalnins), ZentralBlatt

Referenced by:

§28.32(i).

Encodings:

BibTeX

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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External Links:

ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

Encodings:

BibTeX

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Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§29.8.

Encodings:

BibTeX

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Algorithm 911: multiple-precision exponential integral and related functions. ACM Trans. Math. Software 37 (4), pp. Art. 46, 16.

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External Links:

Document, ZentralBlatt

Referenced by:

5th item, §4.48(iii), 3rd item, §6.21(ii).

Encodings:

BibTeX

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Relations among the fundamental solutions of the generalized hypergeometric equation when $p=q+1$. II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.

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External Links:

Document, MathReview (C. S. Meijer), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

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25: 14.30 Spherical and Spheroidal Harmonics

… ► $P_n^m\left(x\right)$ and $Q_n^m\left(x\right)$ ($x>1$) are often referred to as the prolate spheroidal harmonics of the first and second kinds, respectively. … ►

14.30.3 $$Y_{l,m}(\theta ,\varphi )=\frac{{(-1)}^{l+m}}{2^{l}l!}{\left(\frac{(l-m)!(2l+1)}{4\pi (l+m)!}\right)}^{1/2}{\mathrm{e}}^{\mathrm{i}m\varphi }{\left(\sin \theta \right)}^m{\left(\frac{d}{d(\cos \theta )}\right)}^{l+m}{\left(\sin \theta \right)}^{2l}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: 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

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14.30.8 $${\int }_0^{2\pi }{\int }_0^{\pi }\overline{Y_{l_1,m_1}(\theta ,\varphi )}{Y}_{l_2,m_2}(\theta ,\varphi )\sin \theta d\theta d\varphi ={\delta }_{l_1,l_2}{\delta }_{m_1,m_2}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: 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 $${\mathrm{e}}^{t𝐚\cdot 𝐱}=\sqrt{4\pi }\sum _{n=0}^{\mathrm{\infty }}\sum _{m=-n}^n\frac{t^n{r}^n{\lambda }^m{Y}_{n,m}(\theta ,\varphi )}{\sqrt{(2n+1)(n+m)!(n-m)!}},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $Y_{l,m}(\theta ,\varphi )$: 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

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26: 8.22 Mathematical Applications

§8.22 Mathematical Applications

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§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function

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8.22.2 $${\zeta }_x(s)=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^x\frac{t^{s-1}}{{\mathrm{e}}^t-1}dt,$$ $\mathrm{\Re }s>1$,

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Defines:

${\zeta }_x(s)$: incomplete Riemann zeta function (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/8.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.22(ii), §8.22 and Ch.8

►so that ${lim}_{x\to \mathrm{\infty }}{\zeta }_x(s)=\zeta \left(s\right)$, then … ►The Debye functions ${\int }_0^x{t}^n{\left({\mathrm{e}}^t-1\right)}^{-1}dt$ and ${\int }_x^{\mathrm{\infty }}{t}^n{\left({\mathrm{e}}^t-1\right)}^{-1}dt$ are closely related to the incomplete Riemann zeta function and the Riemann zeta function. …

27: 1.17 Integral and Series Representations of the Dirac Delta

§1.17 Integral and Series Representations of the Dirac Delta

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§1.17(ii) Integral Representations

… ►The inner integral does not converge. … ►

Sine and Cosine Functions

… ►Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv). …

28: 8.11 Asymptotic Approximations and Expansions

… ►in both cases uniformly with respect to bounded real values of $y$. For Dawson’s integral $F\left(y\right)$ see §7.2(ii). See Tricomi (1950b) for these approximations, together with higher terms and extensions to complex variables. For related expansions involving Hermite polynomials see Pagurova (1965). … ►As $z\to \mathrm{\infty }$, …

29: 18.17 Integrals

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18.17.34_5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xz}{L}_m^{(\alpha )}\left(x\right)L_n^{(\alpha )}\left(x\right){\mathrm{e}}^{-x}{x}^{\alpha }dx=\frac{\mathrm{\Gamma }\left(\alpha +m+1\right)\mathrm{\Gamma }\left(\alpha +n+1\right)}{\mathrm{\Gamma }\left(\alpha +1\right)m!n!}\frac{z^{m+n}}{{(z+1)}^{\alpha +m+n+1}}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-m,-n}{\alpha +1};z^{-2}),$$ $\mathrm{\Re }z>-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Source:

Erdélyi et al. (1954a, 4.11(35))

Proof sketch:

First assume $z>-1$. Make change of integration variable $x\to x/(z+1)$. Twice substitute (18.18.12) with $\lambda ={(z+1)}^{-1}$. Then use the orthogonality relation for the Laguerre polynomials, see Table 18.3.1. Finally perform analytic continuation with respect to $z$.

Referenced by:

Erratum (V1.2.0) Section 18.17

Permalink:

http://dlmf.nist.gov/18.17.E34_5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vi), §18.17(vi), §18.17 and Ch.18

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30: 18.33 Polynomials Orthogonal on the Unit Circle

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§18.33(ii) Recurrence Relations

… ►For an alternative and more detailed approach to the recurrence relations, see §18.33(vi). … ►

Szegő–Askey

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Recurrence Relations

… ►Equivalent to the recurrence relations (18.33.23), (18.33.24) are the inverse Szegő recurrence relations …