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41: 19.23 Integral Representations

§19.23 Integral Representations

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42: 16.17 Definition

… ►Then the Meijer $G$-function is defined via the Mellin–Barnes integral representation: … ►

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Figure 16.17.1: s-plane. Path $L$ for the integral representation (16.17.1) of the Meijer $G$-function.

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43: 9.13 Generalized Airy Functions

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§9.13(ii) Generalizations from Integral Representations

… ►and the difference equation … ►Connection formulas for the solutions of (9.13.31) include … ►For further generalizations via integral representations see Chin and Hedstrom (1978), Janson et al. (1993, §10), and Kamimoto (1998).

44: 8.19 Generalized Exponential Integral

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§8.19(i) Definition and Integral Representations

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Other Integral Representations

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8.19.4 $$E_p\left(z\right)=\frac{z^{p-1}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(p\right)}{\int }_0^{\mathrm{\infty }}\frac{t^{p-1}{\mathrm{e}}^{-zt}}{1+t}dt,$$ , $\mathrm{\Re }p>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $p$: parameter

Keywords:

Laplace transform, Mellin transform

Permalink:

http://dlmf.nist.gov/8.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19(i), §8.19 and Ch.8

►Integral representations of Mellin–Barnes type for $E_p\left(z\right)$ follow immediately from (8.6.11), (8.6.12), and (8.19.1). …

45: 22.16 Related Functions

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Integral Representation

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Integral Representations

►For , …See Figure 22.16.2. …

46: 8.17 Incomplete Beta Functions

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§8.17(iii) Integral Representation

… ►Further integral representations can be obtained by combining the results given in §8.17(ii) with §15.6. …

47: 25.11 Hurwitz Zeta Function

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§25.11(vii) Integral Representations

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25.11.25 $$\zeta (s,a)=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{1-{\mathrm{e}}^{-x}}dx,$$ $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$.

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Symbols:

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

Keywords:

improper integral, integralrepresentation

Sources:

Srivastava and Choi (2001, (2), p. 89); Apostol (1976, (5), p. 251)

Referenced by:

(25.11.27), (25.11.30), (25.11.35), §25.11(x)

Permalink:

http://dlmf.nist.gov/25.11.E25

Encodings:

pMML, png, TeX

See also:

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

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25.11.26 $$\zeta (s,a)=-s{\int }_{-a}^{\mathrm{\infty }}\frac{x-\lfloor x\rfloor -\frac{1}{2}}{{(x+a)}^{s+1}}dx,$$ , .

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

improper integral, integralrepresentation

Source:

Berndt (1972, (5.3), p. 156)

Permalink:

http://dlmf.nist.gov/25.11.E26

Encodings:

pMML, png, TeX

See also:

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

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§25.11(viii) Further Integral Representations

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25.11.35 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{{(n+a)}^s}=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{1+{\mathrm{e}}^{-x}}dx=2^{-s}\left(\zeta (s,\frac{1}{2}a)-\zeta (s,\frac{1}{2}(1+a))\right),$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }s>0$; or $\mathrm{\Re }a=0$, $\mathrm{\Im }a\ne 0$, .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\int$: integral, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

improper integral, infinite series, integralrepresentation, series representation

Proof sketch:

The infinite series connection to the difference of Hurwitz zeta functions is a restatement of (25.11.8). The integral representation is derivable from the difference of Hurwitz zeta functions by substituting (25.11.25) in each, making a change of variables $x=2t$, factoring, and expressing $1-{\mathrm{e}}^{-2t}=(1+{\mathrm{e}}^{-t})(1-{\mathrm{e}}^{-t})$.

Referenced by:

(25.11.31), §25.11(x), §25.11(x)

Permalink:

http://dlmf.nist.gov/25.11.E35

Encodings:

pMML, png, TeX

See also:

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

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48: 8.21 Generalized Sine and Cosine Integrals

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§8.21(iii) Integral Representations

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8.21.21 $$\mathrm{ci}(a,z)=-f(a,z)\sin z+g(a,z)\cos z.$$

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Symbols:

$\cos z$: cosine function, $\mathrm{ci}(a,z)$: generalized cosine integral, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $f(a,z)$: auxiliary function and $g(a,z)$: auxiliary function

Referenced by:

§8.21(viii)

Permalink:

http://dlmf.nist.gov/8.21.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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49: 13.29 Methods of Computation

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§13.29(iii) Integral Representations

►The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. …

50: 5.13 Integrals

§5.13 Integrals

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