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1: 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(iii) Series Representations

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Legendre Polynomials (§§14.7(i) and 18.3)

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Laguerre Polynomials (§18.3)

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Hermite Polynomials (§18.3)

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2: 25.16 Mathematical Applications

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25.16.1 $$\psi \left(x\right)=\sum _{m=1}^{\mathrm{\infty }}\sum _{p^m\le x}\ln p,$$

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Defines:

$\psi \left(x\right)$: Chebyshev $\psi$-function

Symbols:

$\ln z$: principal branch of logarithm function, $m$: nonnegative integer, $p$: prime number and $x$: real variable

Keywords:

definition, double series, infinite series, seriesrepresentation

Source:

Apostol (1976, p. 75)

Permalink:

http://dlmf.nist.gov/25.16.E1

Encodings:

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See also:

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

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25.16.5 $$H\left(s\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{H_n}{n^s},$$

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Symbols:

$H\left(s\right)$: Euler sums, $H_n$: harmonic number, $n$: nonnegative integer, $s$: complex variable and $h(n)$: harmonic number

Keywords:

Euler sum, infinite series, seriesrepresentation

Source:

Apostol and Vu (1984, p. 85)

Referenced by:

Erratum (V1.1.4) for Notation

Permalink:

http://dlmf.nist.gov/25.16.E5

Encodings:

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Notation (effective with 1.1.4):

The notation previously used for the harmonic number $h(n)$ has been replaced to be $H_n$.

Suggested 2021-08-23 by Gergő Nemes

See also:

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

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25.16.6 $$H\left(s\right)=-{\zeta }^{\prime }\left(s\right)+\gamma \zeta \left(s\right)+\frac{1}{2}\zeta \left(s+1\right)+\sum _{r=1}^k\zeta \left(1-2r\right)\zeta \left(s+2r\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{2k+2}}dx,$$

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Symbols:

$\gamma$: Euler’s constant, $H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler sum, finite sum, improper integral, infinite series, integral representation, seriesrepresentation

Source:

Apostol and Vu (1984, (3), p. 87)

Permalink:

http://dlmf.nist.gov/25.16.E6

Encodings:

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See also:

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

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25.16.7 $$H\left(s\right)=\frac{1}{2}\zeta \left(s+1\right)+\frac{\zeta \left(s\right)}{s-1}-\sum _{r=1}^k\left(\genfrac{}{}{0.0pt}{}{s+2r-2}{2r-1}\right)\zeta \left(1-2r\right)\zeta \left(s+2r\right)-\left(\genfrac{}{}{0.0pt}{}{s+2k}{2k+1}\right)\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{s+2k+1}}dx.$$

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Symbols:

$H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler sum, finite sum, improper integral, infinite series, integral representation, seriesrepresentation

Source:

Apostol and Vu (1984, (6), p. 88)

Permalink:

http://dlmf.nist.gov/25.16.E7

Encodings:

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See also:

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

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25.16.11 $$H(s,z)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}\sum _{m=1}^n\frac{1}{m^z},$$ $\mathrm{\Re }\left(s+z\right)>1$,

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Symbols:

$H(s,z)$: generalized Euler sums, $\mathrm{\Re }$: real part, $m$: nonnegative integer, $n$: nonnegative integer, $s$: complex variable and $z$: complex variable

Keywords:

Euler sum, infinite series, seriesrepresentation

Source:

Apostol and Vu (1984, (1), p. 86)

Permalink:

http://dlmf.nist.gov/25.16.E11

Encodings:

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See also:

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

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3: 25.13 Periodic Zeta Function

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25.13.1 $$F(x,s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}nx}}{n^s},$$

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Defines:

$F(x,s)$: periodic zeta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

definition, infinite series, seriesrepresentation

Source:

Apostol (1976, (9), p. 257)

Permalink:

http://dlmf.nist.gov/25.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

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4: 25.11 Hurwitz Zeta Function

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§25.11(iv) Series Representations

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25.11.8 $$\zeta (s,\frac{1}{2}a)=\zeta (s,\frac{1}{2}a+\frac{1}{2})+2^s\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{{(n+a)}^s},$$ $\mathrm{\Re }s>0$, $s\ne 1$, .

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable

Keywords:

infinite series, seriesrepresentation

Source:

Srivastava and Choi (2001, (5), p. 89)

Referenced by:

(25.11.35), §25.11(x)

Permalink:

http://dlmf.nist.gov/25.11.E8

Encodings:

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See also:

Annotations for §25.11(iv), §25.11 and Ch.25

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§25.11(x) Further Series Representations

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25.11.37 $$\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^k}{k}\zeta (nk,a)=-n\ln \mathrm{\Gamma }\left(a\right)+\ln \left(\prod _{j=0}^{n-1}\mathrm{\Gamma }\left(a-{\mathrm{e}}^{(2j+1)\pi \mathrm{i}/n}\right)\right),$$ $n=2,3,4,\mathrm{\dots }$, $\mathrm{\Re }a\ge 1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer and $a$: real or complex parameter

Keywords:

infinite series, seriesrepresentation

Source:

Adamchik and Srivastava (1998, (2.12), p. 136); with the choice $\mathrm{ph}\left(-1\right)=\pi$, which is reasonable since the gamma function is single valued

Permalink:

http://dlmf.nist.gov/25.11.E37

Encodings:

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See also:

Annotations for §25.11(xi), §25.11 and Ch.25

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25.11.38 $$\sum _{k=1}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{n+k}{k}\right)\zeta (n+k+1,a)z^k=\frac{{(-1)}^n}{n!}\left({\psi }^{(n)}\left(a\right)-{\psi }^{(n)}\left(a-z\right)\right),$$ $n=1,2,3,\mathrm{\dots }$, $\mathrm{\Re }a>0$, .

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter and $z$: complex variable

Keywords:

infinite series, seriesrepresentation

Source:

Adamchik and Srivastava (1998, (2.29), p. 138)

Permalink:

http://dlmf.nist.gov/25.11.E38

Encodings:

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See also:

Annotations for §25.11(xi), §25.11 and Ch.25

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5: 25.12 Polylogarithms

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25.12.7 $${\mathrm{Li}}_2\left({\mathrm{e}}^{\mathrm{i}\theta }\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{\cos \left(n\theta \right)}{n^2}+\mathrm{i}\sum _{n=1}^{\mathrm{\infty }}\frac{\sin \left(n\theta \right)}{n^2}.$$

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Symbols:

$\cos z$: cosine function, ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $n$: nonnegative integer and $\theta$: phase

Keywords:

infinite series, seriesrepresentation

Source:

Maximon (2003, (8.7), p. 2813)

Referenced by:

§25.12(i)

Permalink:

http://dlmf.nist.gov/25.12.E7

Encodings:

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See also:

Annotations for §25.12(i), §25.12 and Ch.25

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25.12.10 $${\mathrm{Li}}_s\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n^s}.$$

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Defines:

${\mathrm{Li}}_s\left(z\right)$: polylogarithm

Symbols:

$n$: nonnegative integer, $s$: complex variable and $z$: complex variable

Keywords:

definition, infinite series, seriesrepresentation

Source:

Lewin (1981, p. 236)

Referenced by:

(25.14.3)

Permalink:

http://dlmf.nist.gov/25.12.E10

Encodings:

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See also:

Annotations for §25.12(ii), §25.12 and Ch.25

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25.12.12 $${\mathrm{Li}}_s\left(z\right)=\mathrm{\Gamma }\left(1-s\right){\left(\ln \frac{1}{z}\right)}^{s-1}+\sum _{n=0}^{\mathrm{\infty }}\zeta \left(s-n\right)\frac{{(\ln z)}^n}{n!},$$ $s\ne 1,2,3,\mathrm{\dots }$, ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $n$: nonnegative integer, $s$: complex variable and $z$: complex variable

Keywords:

infinite series, seriesrepresentation

Source:

Erdélyi et al. (1953a, (1.11.8), p. 29); take $v=1$ and use (25.14.3), (25.11.2)

Permalink:

http://dlmf.nist.gov/25.12.E12

Encodings:

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See also:

Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

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6: 12.18 Methods of Computation

… ►These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …

7: 25.15 Dirichlet $L$-functions

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25.15.1 $$L(s,\chi )=\sum _{n=1}^{\mathrm{\infty }}\frac{\chi (n)}{n^s},$$ $\mathrm{\Re }s>1$,

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Defines:

$L(s,\chi )$: Dirichlet $L$-function

Symbols:

$\mathrm{\Re }$: real part, $n$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

definition, infinite series, seriesrepresentation

Source:

Apostol (1976, p. 249)

Referenced by:

(25.11.36), (25.11.36), §25.11(x), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.15.E1

Encodings:

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See also:

Annotations for §25.15(i), §25.15 and Ch.25

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8: Bibliography V

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Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 72, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1466-2, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions. Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1493-X, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§18.38(iii), Ch.35.

Encodings:

BibTeX

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Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1492-1, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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Fourier series representation of Ferrers function $𝖯$ .

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External Links:

FullText

Referenced by:

(14.13.1).

Encodings:

BibTeX

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9: Bibliography E

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A new series representation for $\zeta (3)$ . Amer. Math. Monthly 97 (3), pp. 219–220.

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External Links:

ISSN 0002-9890, FullText, MathReview (Christian Radoux), ZentralBlatt

Referenced by:

(25.8.10).

Encodings:

BibTeX

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10: 13.12 Products

… ►For integral representations, integrals, and series containing products of $M(a,b,z)$ and $U(a,b,z)$ see Erdélyi et al. (1953a, §6.15.3).