functions Fℓ(η,ρ),Gℓ(η,ρ),H±ℓ(η,ρ)

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21: 27.4 Euler Products and Dirichlet Series

… ►The Riemann zeta function is the prototype of series of the form ►

27.4.4 $$F(s)=\sum _{n=1}^{\mathrm{\infty }}f(n)n^{-s},$$

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Symbols:

$n$: positive integer, $f$: multiplicative function and $F(s)$: generating function

Permalink:

http://dlmf.nist.gov/27.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

… ►The function $F(s)$ is a generating function, or more precisely, a Dirichlet generating function, for the coefficients. …

22: 33.12 Asymptotic Expansions for Large $\eta$

… ►

33.12.2 $$\genfrac{}{}{0pt}{}{F_0(\eta ,\rho )}{G_0(\eta ,\rho )}\sim {\pi }^{1/2}{(2\eta )}^{1/6}\left\{\genfrac{}{}{0pt}{}{\mathrm{Ai}\left(x\right)}{\mathrm{Bi}\left(x\right)}\left(1+\frac{B_1}{\mu }+\frac{B_2}{{\mu }^2}+\mathrm{\cdots }\right)+\genfrac{}{}{0pt}{}{{\mathrm{Ai}}^{\prime }\left(x\right)}{{\mathrm{Bi}}^{\prime }\left(x\right)}\left(\frac{A_1}{\mu }+\frac{A_2}{{\mu }^2}+\mathrm{\cdots }\right)\right\},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable, $\eta$: real parameter, $x$: variable, $\mu$: variable, $A_j$: coefficients and $B_j$: coefficients

A&S Ref:

14.4.9

Permalink:

http://dlmf.nist.gov/33.12.E2

Encodings:

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See also:

Annotations for §33.12(i), §33.12 and Ch.33

►

33.12.3 $$\genfrac{}{}{0pt}{}{F_0^{\prime }(\eta ,\rho )}{G_0^{\prime }(\eta ,\rho )}\sim -{\pi }^{1/2}{(2\eta )}^{-1/6}\left\{\genfrac{}{}{0pt}{}{\mathrm{Ai}\left(x\right)}{\mathrm{Bi}\left(x\right)}\left(\frac{B_1^{\prime }+x{A}_1}{\mu }+\frac{B_2^{\prime }+x{A}_2}{{\mu }^2}+\mathrm{\cdots }\right)+\genfrac{}{}{0pt}{}{{\mathrm{Ai}}^{\prime }\left(x\right)}{{\mathrm{Bi}}^{\prime }\left(x\right)}\left(\frac{B_1+A_1^{\prime }}{\mu }+\frac{B_2+A_2^{\prime }}{{\mu }^2}+\mathrm{\cdots }\right)\right\},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable, $\eta$: real parameter, $x$: variable, $\mu$: variable, $A_j$: coefficients and $B_j$: coefficients

A&S Ref:

14.4.10

Permalink:

http://dlmf.nist.gov/33.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

… ►

33.12.6 $$\genfrac{}{}{0pt}{}{F_0(\eta ,2\eta )}{3^{-1/2}{G}_0(\eta ,2\eta )}\sim \frac{\mathrm{\Gamma }\left(\frac{1}{3}\right){\omega }^{1/2}}{2\sqrt{\pi }}\left(1\mp \frac{2}{35}\frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\frac{1}{{\omega }^4}-\frac{8}{2025}\frac{1}{{\omega }^6}\mp \frac{5792}{\mathrm{46\hspace{0.17em}06875}}\frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\frac{1}{{\omega }^{10}}-\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\eta$: real parameter and $\omega$: factor

Permalink:

http://dlmf.nist.gov/33.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

►

33.12.7 $$\genfrac{}{}{0pt}{}{F_0^{\prime }(\eta ,2\eta )}{3^{-1/2}{G}_0^{\prime }(\eta ,2\eta )}\sim \frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{2\sqrt{\pi }{\omega }^{1/2}}\left(\pm 1+\frac{1}{15}\frac{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\frac{1}{{\omega }^2}\pm \frac{2}{14175}\frac{1}{{\omega }^6}+\frac{1436}{\mathrm{23\hspace{0.17em}38875}}\frac{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\frac{1}{{\omega }^8}\pm \mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\eta$: real parameter and $\omega$: factor

Referenced by:

§33.12(i)

Permalink:

http://dlmf.nist.gov/33.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

…

23: 33.1 Special Notation

… ►The main functions treated in this chapter are first the Coulomb radial functions $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, $c(ϵ,\mathrm{\ell };r)$ (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions. …

24: 8.27 Approximations

… ►

DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}{\mathrm{e}}^{x^2/2}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/2}{t}^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^2-p)+\sqrt{{(x^2-p)}^2+8(x^2+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to \mathrm{\infty }$.

…

25: 10.59 Integrals

§10.59 Integrals

►

10.59.1

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: 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, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind and $n$: integer

A&S Ref:

11.4.26

Referenced by:

§10.59

Permalink:

http://dlmf.nist.gov/10.59.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.59 and Ch.10

… ►For an integral representation of the Dirac delta in terms of a product of spherical Bessel functions of the first kind see §1.17(ii), and for a generalization see Maximon (1991). …

26: 16.4 Argument Unity

… ►The function ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$ is well-poised if … ►The function ${}_{q+1}{}^{}F_q^{}$ with argument unity and general values of the parameters is discussed in Bühring (1992). … ►For generalizations involving ${}_{r+3}{}^{}F_{r+2}^{}$ functions see Kim et al. (2013). … ►The function ${}_3{}^{}F_2^{}(a,b,c;d,e;1)$ is analytic in the parameters $a,b,c,d,e$ when its series expansion converges and the bottom parameters are not negative integers or zero. … ►For continued fractions for ratios of ${}_3{}^{}F_2^{}$ functions with argument unity, see Cuyt et al. (2008, pp. 315–317). …

27: 1.10 Functions of a Complex Variable

… ►Let $F(z)$ be a multivalued function and $D$ be a domain. … ►The function $F(z)={(1-z)}^{\alpha }{(1+z)}^{\beta }$ is many-valued with branch points at $\pm 1$. … ►

1.10.23 $$F(z)=\prod _{n=1}^{\mathrm{\infty }}{a}_n(z),$$ $z\in D$,

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Defines:

$F$: function (locally)

Symbols:

$\in$: element of, $z$: variable, $n$: nonnegative integer and $D$: domain

Permalink:

http://dlmf.nist.gov/1.10.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(x), §1.10 and Ch.1

… ►

1.10.24 $$\frac{F^{\prime }(z)}{F(z)}=\sum _{n=1}^{\mathrm{\infty }}\frac{a_n^{\prime }(z)}{a_n(z)}.$$

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Symbols:

$z$: variable, $n$: nonnegative integer and $F$: function

Permalink:

http://dlmf.nist.gov/1.10.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(x), §1.10 and Ch.1

… ►Then $F(x;z)$ is the generating function for the functions $p_n(x)$, which will automatically have an integral representation …

28: 16.16 Transformations of Variables

… ►

16.16.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\beta +{\beta }^{\prime };x,y)={(1-y)}^{-\alpha }{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha ,\beta }{\beta +{\beta }^{\prime }};\frac{x-y}{1-y}),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function and ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function

Permalink:

http://dlmf.nist.gov/16.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

►

16.16.2 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\beta }^{\prime };x,y)={(1-y)}^{-\alpha }{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha ,\beta }{\gamma };\frac{x}{1-y}),$$

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Symbols:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function and ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function

Permalink:

http://dlmf.nist.gov/16.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

►

16.16.3 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,\alpha ;x,y)={(1-y)}^{-{\beta }^{\prime }}{F}_1(\beta ;\alpha -{\beta }^{\prime },{\beta }^{\prime };\gamma ;x,\frac{x}{1-y}),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function and $F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function

Permalink:

http://dlmf.nist.gov/16.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

… ►

16.16.5 $$F_3(\alpha ,\gamma -\alpha ;\beta ,\gamma -\beta ;\gamma ;x,y)={(1-y)}^{\alpha +\beta -\gamma }{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha ,\beta }{\gamma };x+y-xy),$$

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Symbols:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function and ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function

Permalink:

http://dlmf.nist.gov/16.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

… ►

16.16.7 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x(1-y),y(1-x))=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_k{\left(\beta \right)}_k{\left(\alpha +\beta -\gamma -{\gamma }^{\prime }+1\right)}_k}{{\left(\gamma \right)}_k{\left({\gamma }^{\prime }\right)}_{k}k!}{x}^k{y}^k{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha +k,\beta +k}{\gamma +k};x){}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha +k,\beta +k}{{\gamma }^{\prime }+k};y);$$

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Symbols:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

…

29: 10.39 Relations to Other Functions

… ►

10.39.5 $$I_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }{\mathrm{e}}^{\pm z}}{\mathrm{\Gamma }\left(\nu +1\right)}M(\nu +\frac{1}{2},2\nu +1,\mp 2z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.39

Permalink:

http://dlmf.nist.gov/10.39.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

… ►

10.39.9 $$I_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }}{\mathrm{\Gamma }\left(\nu +1\right)}{}_0{}^{}F_1^{}(-;\nu +1;\frac{1}{4}{z}^2),$$

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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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§9.6(iii)

Permalink:

http://dlmf.nist.gov/10.39.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

►

10.39.10 $$I_{\nu }\left(z\right)={(\frac{1}{2}z)}^{\nu }lim𝐅(\lambda ,\mu ;\nu +1;z^2/(4\lambda \mu )),$$

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.39

Permalink:

http://dlmf.nist.gov/10.39.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

… ►For the functions ${}_0{}^{}F_1^{}$ and $𝐅$ see (16.2.1) and §15.2(i).

30: 33.8 Continued Fractions

… ►

33.8.1 $$\frac{F_{\mathrm{\ell }}^{\prime }}{F_{\mathrm{\ell }}}=S_{\mathrm{\ell }+1}-\frac{R_{\mathrm{\ell }+1}^2}{T_{\mathrm{\ell }+1}-}\frac{R_{\mathrm{\ell }+2}^2}{T_{\mathrm{\ell }+2}-}\mathrm{\cdots }.$$

ⓘ

Symbols:

$F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $T_{\mathrm{\ell }}$: factor

Referenced by:

§33.8, §33.8

Permalink:

http://dlmf.nist.gov/33.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.8 and Ch.33

…