generalized exponential integrals

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31: 18.17 Integrals

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18.17.15 $${\mathrm{e}}^{-x}{L}_n^{(\alpha )}\left(x\right)={\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-y}{L}_n^{(\alpha +\mu )}\left(y\right)\frac{{(y-x)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$ $\mu >0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(iv), §18.17(iv)

Permalink:

http://dlmf.nist.gov/18.17.E15

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

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18.17.28_5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-x}{x}^{\alpha }{L}_n^{(\alpha )}\left(x\right){\mathrm{e}}^{\mathrm{i}xy}dx=\frac{\mathrm{\Gamma }\left(\alpha +n+1\right){(-\mathrm{i}y)}^n}{n!{\left(1-\mathrm{i}y\right)}^{\alpha +n+1}},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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 $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v), Erratum (V1.2.0) Section 18.17

Permalink:

http://dlmf.nist.gov/18.17.E28_5

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

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18.17.34 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xz}{L}_n^{(\alpha )}\left(x\right){\mathrm{e}}^{-x}{x}^{\alpha }dx=\frac{\mathrm{\Gamma }\left(\alpha +n+1\right)z^n}{n!{(z+1)}^{\alpha +n+1}},$$ $\mathrm{\Re }z>-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(vii), §18.17(i), §18.17(v), §18.17(vi), §18.17(viii)

Permalink:

http://dlmf.nist.gov/18.17.E34

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

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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

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See also:

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

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18.17.40 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}{L}_n^{(\alpha )}\left(bx\right)x^{z-1}dx=\frac{\mathrm{\Gamma }\left(z+n\right)}{n!}{(a-b)}^n{a}^{-n-z}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,1+\alpha -z}{1-n-z};\frac{a}{a-b}),$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }z>0$.

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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, $n$: nonnegative integer and $x$: real variable

Keywords:

Laplace transform, Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E40

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

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32: 2.3 Integrals of a Real Variable

… ►(In other words, differentiation of (2.3.8) with respect to the parameter $\lambda$ (or $\mu$) is legitimate.) ►Another extension is to more general factors than the exponential function. … ►In general … ►For the more general integral (2.3.19) we assume, without loss of generality, that the stationary point (if any) is at the left endpoint. … ►For extensions to oscillatory integrals with more general $t$-powers and logarithmic singularities see Wong and Lin (1978) and Sidi (2010). …

33: 18.10 Integral Representations

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18.10.6 $$L_n^{(\alpha )}\left(x^2\right)=\frac{2{(-1)}^n}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +\frac{1}{2}\right)n!}{\int }_0^{\mathrm{\infty }}{\int }_0^{\pi }{(x^2-r^2+2ixr\cos \varphi )}^n{\mathrm{e}}^{-r^2}{r}^{2\alpha +1}{(\sin \varphi )}^{2\alpha }d\varphi dr,$$ $\alpha >-\frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E6

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Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

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18.10.9 $$L_n^{(\alpha )}\left(x\right)=\frac{{\mathrm{e}}^x{x}^{-\frac{1}{2}\alpha }}{n!}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{n+\frac{1}{2}\alpha }{J}_{\alpha }\left(2\sqrt{xt}\right)dt,$$ $\alpha >-1$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $t$: real variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.10.14

Referenced by:

§18.10(iv)

Permalink:

http://dlmf.nist.gov/18.10.E9

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See also:

Annotations for §18.10(iv), §18.10(iv), §18.10 and Ch.18

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34: 1.17 Integral and Series Representations of the Dirac Delta

… ►More generally, assume $\varphi (x)$ is piecewise continuous (§1.4(ii)) when $x\in [-c,c]$ for any finite positive real value of $c$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$. … ►The sum ${\sum }_{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}k(x-a)}$ does not converge, but (1.17.18) can be interpreted as a generalized integral in the sense that …

35: 13.10 Integrals

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13.10.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}𝐌(a,c,kt)dt=\mathrm{\Gamma }\left(b\right)z^{-b}{}_2{}^{}𝐅_1^{}(a,b;c;k/z),$$ $\mathrm{\Re }b>0$, $\mathrm{\Re }z>\max (\mathrm{\Re }k,0)$,

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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, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E3

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See also:

Annotations for §13.10(ii), §13.10 and Ch.13

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13.10.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}U(a,c,t)dt=\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(b-c+1\right)z^{-b}{}_2{}^{}𝐅_1^{}(a,b;a+b-c+1;1-\frac{1}{z}),$$ $\mathrm{\Re }b>\max (\mathrm{\Re }c-1,0)$, $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.10.E7

Encodings:

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See also:

Annotations for §13.10(ii), §13.10 and Ch.13

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36: 18.18 Sums

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18.18.5 $$b_n=\frac{n!}{\mathrm{\Gamma }\left(n+\alpha +1\right)}{\int }_0^{\mathrm{\infty }}f(x)L_n^{(\alpha )}\left(x\right){\mathrm{e}}^{-x}{x}^{\alpha }dx.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $n$: nonnegative integer, $f(z)$: analytic function and $x$: real variable

Referenced by:

§18.18(i)

Permalink:

http://dlmf.nist.gov/18.18.E5

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See also:

Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

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37: 1.13 Differential Equations

… ►(More generally $𝒲\left\{w_1(z),\mathrm{\dots },w_n(z)\right\}=det\left[w_k^{(j-1)}(z)\right]$, where $1\le j,k\le n$.) …(More generally in (1.13.5) for $n$th-order differential equations, $f(z)$ is the coefficient multiplying the $(n-1)$th-order derivative of the solution divided by the coefficient multiplying the $n$th-order derivative of the solution, see Ince (1926, §5.2).) … ►

1.13.13 $$w(z)=W(z)\exp \left(-\frac{1}{2}\int f(z)dz\right)$$

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $z$: variable, $w(z)$: solution, $f(z)$: analytic coefficient and $W(\xi )$: solution

Permalink:

http://dlmf.nist.gov/1.13.E13

Encodings:

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See also:

Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

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1.13.16 $$\eta =\int \exp \left(-\int f(z)dz\right)dz.$$

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Defines:

$\eta$: change of variable (locally)

Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $z$: variable and $f(z)$: analytic coefficient

Permalink:

http://dlmf.nist.gov/1.13.E16

Encodings:

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See also:

Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

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1.13.17 $$\frac{d^{2}w}{{d\eta }^2}+g(z)\exp \left(2\int f(z)dz\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $z$: variable, $w(z)$: solution, $f(z)$: analytic coefficient, $\eta$: change of variable and $g(z)$: analytic coefficient

Permalink:

http://dlmf.nist.gov/1.13.E17

Encodings:

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See also:

Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

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38: Bibliography N

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Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.

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External Links:

ISSN 1452-8630, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§5.11(ii), §5.11(ii).

Encodings:

BibTeX

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Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

Encodings:

BibTeX

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Error bounds and exponential improvement for the asymptotic expansion of the Barnes $G$-function. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2172), pp. 20140534, 14.

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External Links:

ISSN 1364-5021, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

§5.17, §5.17.

Encodings:

BibTeX

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Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal. Proc. Roy. Soc. Edinburgh Sect. A 145 (3), pp. 571–596.

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External Links:

ISSN 0308-2105, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

Encodings:

BibTeX

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The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

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Notes:

In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.

External Links:

ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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39: 13.16 Integral Representations

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13.16.9 $$W_{\kappa ,\mu }\left(z\right)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa +c}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{c-1}{}_2{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}+\mu -\kappa ,\frac{1}{2}-\mu -\kappa }{c};-t)dt,$$ ,

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker 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, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.16.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

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40: 4.2 Definitions

… ►With this definition the general logarithm is given by … ►

§4.2(ii) Logarithms to a General Base $a$

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§4.2(iii) The Exponential Function

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Powers with General Bases

… ►but the general value of ${\mathrm{e}}^z$ is …