integral representations

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1: 10.64 Integral Representations

§10.64 Integral Representations

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2: 18.10 Integral Representations

§18.10 Integral Representations

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Ultraspherical

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Legendre

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Jacobi

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Ultraspherical

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3: 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. …

4: 16.25 Methods of Computation

… ►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …

5: 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).

6: 14.25 Integral Representations

§14.25 Integral Representations

… ►For corresponding contour integrals, with less restrictions on $\mu$ and $\nu$, see Olver (1997b, pp. 174–179), and for further integral representations see Magnus et al. (1966, §4.6.1).

7: 25.5 Integral Representations

§25.5 Integral Representations

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25.5.2 $$\zeta \left(s\right)=\frac{1}{\mathrm{\Gamma }\left(s+1\right)}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^x{x}^s}{{({\mathrm{e}}^x-1)}^2}dx,$$ $\mathrm{\Re }s>1$.

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Symbols:

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

Keywords:

improper integral, integralrepresentation

Proof sketch:

Derivable from (25.5.1) by integration by parts.

Permalink:

http://dlmf.nist.gov/25.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(i), §25.5 and Ch.25

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25.5.5 $$\zeta \left(s\right)=-s{\int }_0^{\mathrm{\infty }}\frac{x-\lfloor x\rfloor -\frac{1}{2}}{x^{s+1}}dx,$$ .

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integralrepresentation

Source:

Titchmarsh (1986b, (2.1.6), p. 15)

Permalink:

http://dlmf.nist.gov/25.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(i), §25.5 and Ch.25

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25.5.19 $$\zeta \left(m+s\right)={(-1)}^{m-1}\frac{\mathrm{\Gamma }\left(s\right)\sin \left(\pi s\right)}{\pi \mathrm{\Gamma }\left(m+s\right)}{\int }_0^{\mathrm{\infty }}{\psi }^{(m)}\left(1+x\right)x^{-s}dx,$$ $m=1,2,3,\mathrm{\dots }$.

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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, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\sin z$: sine function, $m$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integralrepresentation

Source:

de Bruijn (1937, p. 178)

Referenced by:

§25.5(ii)

Permalink:

http://dlmf.nist.gov/25.5.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(ii), §25.5 and Ch.25

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§25.5(iii) Contour Integrals

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8: 24.7 Integral Representations

§24.7 Integral Representations

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§24.7(i) Bernoulli and Euler Numbers

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24.7.5 $$B_{2n}={(-1)}^n\frac{2n(2n-1)}{\pi }{\int }_0^{\mathrm{\infty }}{t}^{2n-2}\ln \left(1-{\mathrm{e}}^{-2\pi t}\right)dt.$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function, $n$: integer and $t$: real or complex

Referenced by:

§24.7(i)

Permalink:

http://dlmf.nist.gov/24.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(i), §24.7 and Ch.24

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§24.7(ii) Bernoulli and Euler Polynomials

… ►For further integral representations see Prudnikov et al. (1986a, §§2.3–2.6) and Gradshteyn and Ryzhik (2015, Chapters 3 and 4).

9: 13.25 Products

… ►For integral representations, integrals, and series containing products of $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ see Erdélyi et al. (1953a, §6.15.3).

10: 23.11 Integral Representations

§23.11 Integral Representations

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