Euler integral

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21: 8.8 Recurrence Relations and Derivatives

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8.8.14 $${\frac{\partial }{\partial a}{\gamma }^{\ast }(a,z)|}_{a=0}=-E_1\left(z\right)-\ln z.$$

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Symbols:

$E_1\left(z\right)$: exponential integral, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $a$: parameter

A&S Ref:

6.5.24

Permalink:

http://dlmf.nist.gov/8.8.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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22: 13.6 Relations to Other Functions

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13.6.6 $$U(a,a,z)=z^{1-a}U(1,2-a,z)=z^{1-a}{\mathrm{e}}^z{E}_a\left(z\right)={\mathrm{e}}^z\mathrm{\Gamma }(1-a,z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

A&S Ref:

13.6.28 (in different form)

Permalink:

http://dlmf.nist.gov/13.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(ii), §13.6 and Ch.13

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23: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.5 $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(c-a\right)>\frac{1}{2}(m-1)$, .

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $c$: complex variable, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.7(iv)

Permalink:

http://dlmf.nist.gov/35.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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24: 25.5 Integral Representations

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25.5.1 $$\zeta \left(s\right)=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}}{{\mathrm{e}}^x-1}dx,$$ $\mathrm{\Re }s>1$.

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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, integral representation

Source:

Erdélyi et al. (1953a, (1.12.4), p. 32)

A&S Ref:

23.2.7

Referenced by:

§25.12(ii), (25.5.2), (25.5.6)

Permalink:

http://dlmf.nist.gov/25.5.E1

Encodings:

pMML, png, TeX

See also:

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

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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, integral representation

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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25.5.3 $$\zeta \left(s\right)=\frac{1}{(1-2^{1-s})\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}}{{\mathrm{e}}^x+1}dx,$$ $\mathrm{\Re }s>0$.

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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, integral representation

Source:

Erdélyi et al. (1953a, (1.12.5), p. 32)

A&S Ref:

23.2.8

Referenced by:

(25.5.4)

Permalink:

http://dlmf.nist.gov/25.5.E3

Encodings:

pMML, png, TeX

See also:

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

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25.5.6 $$\zeta \left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{{\mathrm{e}}^x}dx,$$ $\mathrm{\Re }s>-1$.

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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, integral representation

Proof sketch:

Derivable from (25.5.1) by first replacing $x^{s-1}\mapsto x^{s-1}(1+{\mathrm{e}}^{-x}-{\mathrm{e}}^{-x})$, using the identity ${\mathrm{e}}^{-x}=(1-{\mathrm{e}}^{-x})/({\mathrm{e}}^x-1)$, and replacing ${({\mathrm{e}}^x-1)}^{-1}\mapsto ({({\mathrm{e}}^x-1)}^{-1}-1/x+1/2-1/2+1/x)$ in the resulting integrand with (5.2.1), (5.5.1).

Referenced by:

(25.5.7)

Permalink:

http://dlmf.nist.gov/25.5.E6

Encodings:

pMML, png, TeX

See also:

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

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25.5.8 $$\zeta \left(s\right)=\frac{1}{2(1-2^{-s})\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}}{\sinh x}dx,$$ $\mathrm{\Re }s>1$.

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Symbols:

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

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.12.6), p. 32)

Permalink:

http://dlmf.nist.gov/25.5.E8

Encodings:

pMML, png, TeX

See also:

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

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25: 6.10 Other Series Expansions

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6.10.6 $$\mathrm{Ei}\left(x\right)=\gamma +\ln \left|x\right|+\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n(x-a_n){\left({𝗂}_n^{(1)}\left(\frac{1}{2}x\right)\right)}^2,$$ $x\ne 0$,

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $x$: real variable, $n$: nonnegative integer and $a_n$: coefficients

Referenced by:

§6.18(i)

Permalink:

http://dlmf.nist.gov/6.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

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26: 8.19 Generalized Exponential Integral

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8.19.1 $$E_p\left(z\right)=z^{p-1}\mathrm{\Gamma }(1-p,z).$$

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Defines:

$E_p\left(z\right)$: generalized exponential integral

Symbols:

$\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $p$: parameter

A&S Ref:

5.1.45 6.5.9 (Definition extended to general values of $p$.)

Referenced by:

§8.19(i), §8.19(iii), §8.19(iv), §8.19(v), §8.19(vi), §8.19(vii)

Permalink:

http://dlmf.nist.gov/8.19.E1

Encodings:

pMML, png, TeX

See also:

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

►Most properties of $E_p\left(z\right)$ follow straightforwardly from those of $\mathrm{\Gamma }(a,z)$. … ►

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

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8.19.10 $$E_p\left(z\right)=z^{p-1}\mathrm{\Gamma }\left(1-p\right)-\sum _{k=0}^{\mathrm{\infty }}\frac{{(-z)}^k}{k!(1-p+k)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $E_p\left(z\right)$: generalized exponential integral, $!$: factorial (as in $n!$), $z$: complex variable, $p$: parameter and $k$: nonnegative integer

Referenced by:

§8.19(iv)

Permalink:

http://dlmf.nist.gov/8.19.E10

Encodings:

pMML, png, TeX

See also:

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

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8.19.11 $$E_p\left(z\right)=\mathrm{\Gamma }\left(1-p\right)\left(z^{p-1}-{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{\mathrm{\Gamma }\left(2-p+k\right)}\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable, $p$: parameter and $k$: nonnegative integer

Referenced by:

§8.19(iv)

Permalink:

http://dlmf.nist.gov/8.19.E11

Encodings:

pMML, png, TeX

See also:

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

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27: 6.4 Analytic Continuation

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6.4.1 $$E_1\left(z\right)=\mathrm{Ein}\left(z\right)-\mathrm{Ln}z-\gamma ;$$

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $E_1\left(z\right)$: exponential integral, $\mathrm{Ln}z$: general logarithm function and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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6.4.3 $$E_1\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\mathrm{Ein}\left(-z\right)-\ln z-\gamma \mp \pi \mathrm{i},$$ $|\mathrm{ph}z|\le \pi$.

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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28: 25.12 Polylogarithms

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25.12.11 $${\mathrm{Li}}_s\left(z\right)\equiv \frac{z}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}}{{\mathrm{e}}^x-z}dx,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\int$: integral, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $x$: real variable, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Lewin (1981, (7.188), p. 236)

Referenced by:

(25.12.16), §25.12(ii)

Permalink:

http://dlmf.nist.gov/25.12.E11

Encodings:

pMML, png, TeX

See also:

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

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25.12.14 $$F_s(x)=\frac{1}{\mathrm{\Gamma }\left(s+1\right)}{\int }_0^{\mathrm{\infty }}\frac{t^s}{{\mathrm{e}}^{t-x}+1}dt,$$ $s>-1$,

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Defines:

$F_s(x)$: Fermi–Dirac integral (locally)

Symbols:

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

Keywords:

definition, Fermi–Dirac integral, improper integral, integral representation

Source:

Dingle (1957b, (1), p. 226)

Referenced by:

(25.12.16), 3rd item, 5th item

Permalink:

http://dlmf.nist.gov/25.12.E14

Encodings:

pMML, png, TeX

See also:

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

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25.12.15 $$G_s(x)=\frac{1}{\mathrm{\Gamma }\left(s+1\right)}{\int }_0^{\mathrm{\infty }}\frac{t^s}{{\mathrm{e}}^{t-x}-1}dt,$$ $s>-1$, ; or $s>0$, $x\le 0$,

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Defines:

$G_s(x)$: Bose–Einstein integral (locally)

Symbols:

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

Keywords:

Bose–Einstein integral, definition, improper integral, integral representation

Source:

Dingle (1957a, p. 240)

Referenced by:

(25.12.16)

Permalink:

http://dlmf.nist.gov/25.12.E15

Encodings:

pMML, png, TeX

See also:

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

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29: 29.18 Mathematical Applications

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$0\le \gamma \le 4K,$

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$\beta =K+\mathrm{i}{\beta }^{\prime },$ $0\le {\beta }^{\prime }\le 2K^{\prime },0\le \gamma \le 4K$,

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30: 13.4 Integral Representations

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13.4.1 $$𝐌(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b-a\right)}{\int }_0^1{\mathrm{e}}^{zt}{t}^{a-1}{(1-t)}^{b-a-1}dt,$$ $\mathrm{\Re }b>\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

13.2.1 (in different form)

Referenced by:

§13.10(v), §13.29(iii)

Permalink:

http://dlmf.nist.gov/13.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

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13.4.3 $$𝐌(a,b,-z)=\frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\mathrm{\Gamma }\left(a\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{a-\frac{1}{2}b-\frac{1}{2}}{J}_{b-1}\left(2\sqrt{zt}\right)dt,$$ $\mathrm{\Re }a>0$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§13.10(v)

Permalink:

http://dlmf.nist.gov/13.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

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13.4.4 $$U(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(a\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{a-1}{(1+t)}^{b-a-1}dt,$$ $\mathrm{\Re }a>0$, ,

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Symbols:

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

A&S Ref:

13.2.5

Referenced by:

§13.10(v), §13.29(iii), §7.11, (9.5.8)

Permalink:

http://dlmf.nist.gov/13.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

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13.4.9 $$𝐌(a,b,z)=\frac{\mathrm{\Gamma }\left(1+a-b\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(a\right)}{\int }_0^{(1+)}{\mathrm{e}}^{zt}{t}^{a-1}{(t-1)}^{b-a-1}dt,$$ $b-a\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }a>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(ii), §13.4 and Ch.13

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13.4.16 $$𝐌(a,b,-z)=\frac{1}{2\pi \mathrm{i}\mathrm{\Gamma }\left(a\right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(-t\right)}{\mathrm{\Gamma }\left(b+t\right)}{z}^{t}dt,$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Keywords:

Mellin transform

A&S Ref:

13.2.9 (in different form)

Permalink:

http://dlmf.nist.gov/13.4.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(iii), §13.4 and Ch.13

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