functions Fℓ(η,ρ),Gℓ(η,ρ),H±ℓ(η,ρ)

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1: 8.22 Mathematical Applications

… ►The so-called terminant function $F_p\left(z\right)$, defined by ►

8.22.1 $$F_p\left(z\right)=\frac{\mathrm{\Gamma }\left(p\right)}{2\pi }{z}^{1-p}{E}_p\left(z\right)=\frac{\mathrm{\Gamma }\left(p\right)}{2\pi }\mathrm{\Gamma }(1-p,z),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $F_p\left(z\right)$: terminant function, $z$: complex variable and $p$: parameter

Permalink:

http://dlmf.nist.gov/8.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.22(i), §8.22 and Ch.8

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2: 2.11 Remainder Terms; Stokes Phenomenon

… ►

2.11.11 $$F_{n+p}\left(z\right)=\frac{{\mathrm{e}}^{-z}}{2\pi }{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}{t}^{n+p-1}}{1+t}dt=\frac{\mathrm{\Gamma }\left(n+p\right)}{2\pi }\frac{E_{n+p}\left(z\right)}{z^{n+p-1}}.$$

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Defines:

$F_p\left(z\right)$: terminant function

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 and $\int$: integral

Permalink:

http://dlmf.nist.gov/2.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

… ►Owing to the factor ${\mathrm{e}}^{-\rho }$, that is, ${\mathrm{e}}^{-|z|}$ in (2.11.13), $F_{n+p}\left(z\right)$ is uniformly exponentially small compared with $E_p\left(z\right)$. … ►However, to enjoy the resurgence property (§2.7(ii)) we often seek instead expansions in terms of the $F$-functions introduced in §2.11(iii), leaving the connection of the error-function type behavior as an implicit consequence of this property of the $F$-functions. In this context the $F$-functions are called terminants, a name introduced by Dingle (1973). … ►

2.11.20 $$R_n^{(1)}(z)={(-1)}^{n-1}\mathrm{i}{\mathrm{e}}^{({\mu }_2-{\mu }_1)\pi \mathrm{i}}{\mathrm{e}}^{{\lambda }_{2}z}{z}^{{\mu }_2}\left(C_1\sum _{s=0}^{m-1}{(-1)}^s{a}_{s,2}\frac{F_{n+{\mu }_2-{\mu }_1-s}\left(z\right)}{z^s}+R_{m,n}^{(1)}(z)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F_p\left(z\right)$: terminant function, $R_n^{(j)}(z)$: remainder, $a_{s,j}$: coefficients, ${\lambda }_j$: roots, ${\mu }_j$: quantities and $C_1$, $C_2$: Stokes multipliers

Referenced by:

§2.11(v)

Permalink:

http://dlmf.nist.gov/2.11.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(v), §2.11 and Ch.2

…

3: 15.1 Special Notation

… ►

15.1.1 $${}_2{}^{}F_1^{}(a,b;c;z)=F(a,b;c;z)=F(\genfrac{}{}{0pt}{}{a,b}{c};z),$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.1.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.1 and Ch.15

… ►

15.1.2 $$\frac{F(a,b;c;z)}{\mathrm{\Gamma }\left(c\right)}=𝐅(a,b;c;z)=𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)={}_2{}^{}𝐅_1^{}(a,b;c;z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.1.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.1 and Ch.15

…

4: 32.14 Combinatorics

… ►

32.14.1 $$\underset{N\to \mathrm{\infty }}{lim}\mathrm{Prob}\left(\frac{{\mathrm{\ell }}_N(𝝅)-2\sqrt{N}}{N^{1/6}}\le s\right)=F(s),$$

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Symbols:

$𝝅(m)$: permutations, ${\mathrm{\ell }}_N(𝝅)$: length and $F(s)$: distribution function

Permalink:

http://dlmf.nist.gov/32.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.14 and Ch.32

►where the distribution function $F(s)$ is defined here by ►

32.14.2 $$F(s)=\exp \left(-{\int }_s^{\mathrm{\infty }}(x-s)w^2(x)dx\right),$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $x$: real and $F(s)$: distribution function

Referenced by:

§32.14

Permalink:

http://dlmf.nist.gov/32.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §32.14 and Ch.32

… ►The distribution function $F(s)$ given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of $n\times n$ Hermitian matrices; see Tracy and Widom (1994). …

5: 16.1 Special Notation

… ►The main functions treated in this chapter are the generalized hypergeometric function ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$, the Appell (two-variable hypergeometric) functions $F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$, $F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$, and the Meijer $G$-function $G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$. Alternative notations are ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$, and ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ for the generalized hypergeometric function, $F_1(\alpha ,\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_2(\alpha ,\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$, $F_3(\alpha ,{\alpha }^{\prime },\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$, for the Appell functions, and $G_{p,q}^{m,n}(z;𝐚;𝐛)$ for the Meijer $G$-function.

6: 33.3 Graphics

… ►

§33.3(i) Line Graphs of the Coulomb Radial Functions $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$

… ►

33.3.1 $$M_{\mathrm{\ell }}(\eta ,\rho )={(F_{\mathrm{\ell }}^2(\eta ,\rho )+G_{\mathrm{\ell }}^2(\eta ,\rho ))}^{1/2}=\left|H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\right|.$$

ⓘ

Defines:

$M_{\mathrm{\ell }}(\eta ,\rho )$: envelope of Coulomb functions

Symbols:

$G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.3(i), §33.3 and Ch.33

… ►

§33.3(ii) Surfaces of the Coulomb Radial Functions $F_0(\eta ,\rho )$ and $G_0(\eta ,\rho )$

…

7: 8.14 Integrals

… ►

8.14.5 $${\int }_0^{\mathrm{\infty }}{x}^{a-1}{\mathrm{e}}^{-sx}\gamma (b,x)dx=\frac{\mathrm{\Gamma }\left(a+b\right)}{b{(1+s)}^{a+b}}F(1,a+b;1+b;1/(1+s)),$$ $\mathrm{\Re }s>0$, $\mathrm{\Re }\left(a+b\right)>0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.14 and Ch.8

►

8.14.6 $${\int }_0^{\mathrm{\infty }}{x}^{a-1}{\mathrm{e}}^{-sx}\mathrm{\Gamma }(b,x)dx=\frac{\mathrm{\Gamma }\left(a+b\right)}{a{(1+s)}^{a+b}}F(1,a+b;1+a;s/(1+s)),$$ $\mathrm{\Re }s>-1$, $\mathrm{\Re }\left(a+b\right)>0$, $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/8.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.14 and Ch.8

►For the hypergeometric function $F(a,b;c;z)$ see §15.2(i). …

8: 16.2 Definition and Analytic Properties

… ►

§16.2(i) Generalized Hypergeometric Series

… ►Equivalently, the function is denoted by ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$ or ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$, and sometimes, for brevity, by ${}_p{}^{}F_q^{}\left(z\right)$. … ►

16.2.3 $${}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{-m,𝐚}{𝐛};z)=\frac{{\left(𝐚\right)}_m{(-z)}^m}{{\left(𝐛\right)}_m}{}_{q+1}{}^{}F_p^{}(\genfrac{}{}{0pt}{}{-m,1-m-𝐛}{1-m-𝐚};\frac{{(-1)}^{p+q}}{z})$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ 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, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $p$: nonnegative integer, $q$: nonnegative integer and $z$: complex variable

Referenced by:

§13.6(v)

Permalink:

http://dlmf.nist.gov/16.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.2(iv), §16.2(iv), §16.2 and Ch.16

… ►

16.2.5 $${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)={}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)/\left(\mathrm{\Gamma }\left(b_1\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_q\right)\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a_1\right)}_k\mathrm{\cdots }{\left(a_p\right)}_k}{\mathrm{\Gamma }\left(b_1+k\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_q+k\right)}\frac{z^k}{k!};$$

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Defines:

${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function

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, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: 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

Referenced by:

§16.2(v)

Permalink:

http://dlmf.nist.gov/16.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §16.2(v), §16.2 and Ch.16

… ►When $p\le q+1$ and $z$ is fixed and not a branch point, any branch of ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ is an entire function of each of the parameters $a_1,\mathrm{\dots },a_p,b_1,\mathrm{\dots },b_q$.

9: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

… ►

§33.5(i) Small $\rho$

… ►

§33.5(ii) $\eta =0$

… ►For the functions $𝗃$, $𝗒$, $J$, $Y$ see §§10.47(ii), 10.2(ii). … ►

§33.5(iii) Small $|\eta |$

… ►

§33.5(iv) Large $\mathrm{\ell }$

…

10: 33.11 Asymptotic Expansions for Large $\rho$

§33.11 Asymptotic Expansions for Large $\rho$

… ►

$F_{\mathrm{\ell }}(\eta ,\rho )=g(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}+f(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$

… ►

$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )=\widehat{g}(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}+\widehat{f}(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$

… ►

33.11.4 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )={\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}}(f(\eta ,\rho )\pm \mathrm{i}g(\eta ,\rho )),$$

ⓘ

Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $f(\eta ,\rho )$: function and $g(\eta ,\rho )$: function

Permalink:

http://dlmf.nist.gov/33.11.E4

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.22):

The arguments of some of the functions in these equations were included to improve clarity of the presentation.

See also:

Annotations for §33.11 and Ch.33

… ►

33.11.7 $$g(\eta ,\rho )\widehat{f}(\eta ,\rho )-f(\eta ,\rho )\widehat{g}(\eta ,\rho )=1.$$

ⓘ

Symbols:

$\rho$: nonnegative real variable, $\eta$: real parameter, $f(\eta ,\rho )$: function, $g(\eta ,\rho )$: function, $\widehat{f}(\eta ,\rho )$: function and $\widehat{g}(\eta ,\rho )$: function

A&S Ref:

14.5.8

Referenced by:

§33.11, Erratum (V1.0.22) for Equations (33.11.2)–(33.11.7), Erratum (V1.0.22) for Equations (33.11.2)–(33.11.7)

Permalink:

http://dlmf.nist.gov/33.11.E7

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.22):

The arguments of some of the functions in this equation were included to improve clarity of the presentation.

See also:

Annotations for §33.11 and Ch.33

…