general base

SearchAdvancedHelp

(0.003 seconds)

21—30 of 165 matching pages

21: 6 Exponential, Logarithmic, Sine, and
Cosine Integrals

…

22: 35.9 Applications

… ►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument ${}_p{}^{}F_q^{}$, with $p\le 2$ and $q\le 1$. …

23: 18.17 Integrals

… ►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

►

18.17.34_5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xz}{L}_m^{(\alpha )}\left(x\right)L_n^{(\alpha )}\left(x\right){\mathrm{e}}^{-x}{x}^{\alpha }dx=\frac{\mathrm{\Gamma }\left(\alpha +m+1\right)\mathrm{\Gamma }\left(\alpha +n+1\right)}{\mathrm{\Gamma }\left(\alpha +1\right)m!n!}\frac{z^{m+n}}{{(z+1)}^{\alpha +m+n+1}}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-m,-n}{\alpha +1};z^{-2}),$$ $\mathrm{\Re }z>-1$.

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

…

24: 16.5 Integral Representations and Integrals

… ►

16.5.3 $${}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)=\frac{1}{\mathrm{\Gamma }\left(a_0\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{a_0-1}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\cdots },b_q};zt)dt,$$ , $\mathrm{\Re }{a}_0>0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Keywords:

Laplace transform, Mellin transform

Referenced by:

§16.5

Permalink:

http://dlmf.nist.gov/16.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.5 and Ch.16

►

16.5.4 $${}_p{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_0,\mathrm{\dots },b_q};z)=\frac{\mathrm{\Gamma }\left(b_0\right)}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^t{t}^{-b_0}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};\frac{z}{t})dt,$$ $c>0$, $\mathrm{\Re }{b}_0>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized 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, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Keywords:

Mellin transform

Referenced by:

§16.5, §16.5

Permalink:

http://dlmf.nist.gov/16.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.5 and Ch.16

…

25: 18.10 Integral Representations

… ►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

…

26: 13.6 Relations to Other Functions

… ►

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

ⓘ

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

…

27: 10.74 Methods of Computation

… ►For evaluation of the Hankel functions $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ for complex values of $\nu$ and $z$ based on the integral representations (10.9.18) see Remenets (1973). ►For applications of generalized Gauss–Laguerre quadrature (§3.5(v)) to the evaluation of the modified Bessel functions $K_{\nu }\left(z\right)$ for and see Gautschi (2002a). The integral representation used is based on (10.32.8). … ►Then $J_n\left(x\right)$ and $Y_n\left(x\right)$ can be generated by either forward or backward recurrence on $n$ when , but if $n>x$ then to maintain stability $J_n\left(x\right)$ has to be generated by backward recurrence on $n$, and $Y_n\left(x\right)$ has to be generated by forward recurrence on $n$. …

28: 14.2 Differential Equations

… ►

14.2.10 $$𝒲\left\{P_{\nu }^{\mu }\left(x\right),Q_{\nu }^{\mu }\left(x\right)\right\}=-{\mathrm{e}}^{\mu \pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +1\right)\left(x^2-1\right)},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.1.8 (modified)

Permalink:

http://dlmf.nist.gov/14.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §14.2(iv), §14.2 and Ch.14

►

14.2.11 $$P_{\nu +1}^{\mu }\left(x\right)Q_{\nu }^{\mu }\left(x\right)-P_{\nu }^{\mu }\left(x\right)Q_{\nu +1}^{\mu }\left(x\right)={\mathrm{e}}^{\mu \pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +2\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.2(iv), §14.2 and Ch.14

29: 14.3 Definitions and Hypergeometric Representations

… ►

14.3.7 $\mu +\nu \ne -1,-2,-3,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and : real variable

A&S Ref:

8.1.3 (modified)

Permalink:

http://dlmf.nist.gov/14.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

… ►

14.3.10

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.20(i), §14.21(i), §14.21(iii), §14.6(i)

Permalink:

http://dlmf.nist.gov/14.3.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

…

30: 9.13 Generalized Airy Functions

… ►

9.13.9 $$A_n\left(z\right)=\sqrt{p/\pi }\sin \left(p\pi \right)z^{-n/4}{\mathrm{e}}^{-\zeta }\left(1+O\left({\zeta }^{-1}\right)\right),$$ $|\mathrm{ph}z|\le 3p\pi -\delta$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $A_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $\sin z$: sine function, $z$: complex variable, $\delta$: small positive constant, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (13))

Permalink:

http://dlmf.nist.gov/9.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

►

9.13.10

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $A_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $z$: complex variable, $\delta$: small positive constant, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (14)–(15))

Permalink:

http://dlmf.nist.gov/9.13.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

►

9.13.11 $$B_n\left(z\right)={\pi }^{-1/2}{z}^{-n/4}{\mathrm{e}}^{\zeta }\left(1+O\left({\zeta }^{-1}\right)\right),$$ $|\mathrm{ph}z|\le p\pi -\delta$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $B_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $z$: complex variable, $\delta$: small positive constant, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (16))

Permalink:

http://dlmf.nist.gov/9.13.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

►

9.13.12

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $B_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\sin z$: sine function, $z$: complex variable, $\delta$: small positive constant, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (17)–(18))

Permalink:

http://dlmf.nist.gov/9.13.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

… ►

9.13.18 $$w=U_m(t{\mathrm{e}}^{-2j\pi \mathrm{i}/m}),$$ $j=0,\pm 1,\pm 2,\mathrm{\dots }.$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U_{\mathrm{i}}$: Smirnov’s generalized Airy function, $w$: function, $m$: index and $j$: index

Notes:

See Olver (1978).

Permalink:

http://dlmf.nist.gov/9.13.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

…