Olver hypergeometric function

SearchAdvancedHelp

(0.012 seconds)

21—30 of 69 matching pages

21: 14.1 Special Notation

… ► ►►►

$x$, $y$, $\tau$	real variables.
…
$𝐅(a,b;c;z)$	Olver’s scaled hypergeometric function: $F(a,b;c;z)/\mathrm{\Gamma }\left(c\right)$.

…

22: 14.23 Values on the Cut

… ►

14.23.3 $${𝑸}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)=\frac{{\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}{\pi }^{3/2}{\left(1-x^2\right)}^{\mu /2}}{2^{\nu +1}}\left(\frac{x𝐅(\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2},\frac{1}{2}\nu +\frac{1}{2}\mu +1;\frac{3}{2};x^2)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{1}{2}\right)}\mp \mathrm{i}\frac{𝐅(\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{1}{2};\frac{1}{2};x^2)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +1\right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)}\right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: 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, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

…

23: 10.16 Relations to Other Functions

… ►

10.16.10 $$J_{\nu }\left(z\right)={(\frac{1}{2}z)}^{\nu }lim𝐅(\lambda ,\mu ;\nu +1;-z^2/(4\lambda \mu )),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: 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

A&S Ref:

9.1.70 (modified)

Referenced by:

§10.39

Permalink:

http://dlmf.nist.gov/10.16.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

…

24: 16.2 Definition and Analytic Properties

… ►

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

ⓘ

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

…

25: 10.22 Integrals

… ►

10.22.49 $${\int }_0^{\mathrm{\infty }}{t}^{\mu -1}{\mathrm{e}}^{-at}{J}_{\nu }\left(bt\right)dt=\frac{{(\frac{1}{2}b)}^{\nu }}{a^{\mu +\nu }}\mathrm{\Gamma }\left(\mu +\nu \right)𝐅(\frac{\mu +\nu }{2},\frac{\mu +\nu +1}{2};\nu +1;-\frac{b^2}{a^2}),$$ $\mathrm{\Re }\left(\mu +\nu \right)>0,\mathrm{\Re }\left(a\pm \mathrm{i}b\right)>0$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Keywords:

Laplace transform, Mellin transform

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22 and Ch.10

►

10.22.50 $${\int }_0^{\mathrm{\infty }}{t}^{\mu -1}{\mathrm{e}}^{-at}{Y}_{\nu }\left(bt\right)dt=\cot \left(\nu \pi \right)\frac{{(\frac{1}{2}b)}^{\nu }\mathrm{\Gamma }\left(\mu +\nu \right)}{{(a^2+b^2)}^{\frac{1}{2}(\mu +\nu )}}𝐅(\frac{\mu +\nu }{2},\frac{1-\mu +\nu }{2};\nu +1;\frac{b^2}{a^2+b^2})-\csc \left(\nu \pi \right)\frac{{(\frac{1}{2}b)}^{-\nu }\mathrm{\Gamma }\left(\mu -\nu \right)}{{(a^2+b^2)}^{\frac{1}{2}(\mu -\nu )}}𝐅(\frac{\mu -\nu }{2},\frac{1-\mu -\nu }{2};1-\nu ;\frac{b^2}{a^2+b^2}),$$ $\mathrm{\Re }\mu >|\mathrm{\Re }\nu |,\mathrm{\Re }\left(a\pm \mathrm{i}b\right)>0$.

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.22.E50

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22 and Ch.10

… ►

10.22.56 $${\int }_0^{\mathrm{\infty }}\frac{J_{\mu }\left(at\right)J_{\nu }\left(bt\right)}{t^{\lambda }}dt=\frac{a^{\mu }\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu -\frac{1}{2}\lambda +\frac{1}{2}\right)}{2^{\lambda }{b}^{\mu -\lambda +1}\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\lambda +\frac{1}{2}\right)}𝐅(\frac{1}{2}(\mu +\nu -\lambda +1),\frac{1}{2}(\mu -\nu -\lambda +1);\mu +1;\frac{a^2}{b^2}),$$ , $\mathrm{\Re }\left(\mu +\nu +1\right)>\mathrm{\Re }\lambda >-1$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Keywords:

Mellin transform

A&S Ref:

11.4.33, 11.4.34

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E56

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

… ►

10.22.58 $${\int }_0^{\mathrm{\infty }}\frac{J_{\nu }\left(at\right)J_{\nu }\left(bt\right)}{t^{\lambda }}dt=\frac{{(ab)}^{\nu }\mathrm{\Gamma }\left(\nu -\frac{1}{2}\lambda +\frac{1}{2}\right)}{2^{\lambda }{(a^2+b^2)}^{\nu -\frac{1}{2}\lambda +\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}\lambda +\frac{1}{2}\right)}𝐅(\frac{2\nu +1-\lambda }{4},\frac{2\nu +3-\lambda }{4};\nu +1;\frac{4a^2{b}^2}{{(a^2+b^2)}^2}),$$ $a\ne b$, $\mathrm{\Re }\left(2\nu +1\right)>\mathrm{\Re }\lambda >-1$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/10.22.E58

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

… ►

10.22.64

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $n$: integer

Referenced by:

§10.22(iv), §10.59

Permalink:

http://dlmf.nist.gov/10.22.E64

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

…

26: 13.8 Asymptotic Approximations for Large Parameters

… ►

13.8.12 $$𝐌(a,b,z)\sim {\left(z/a\right)}^{(1-b)/2}\frac{{\mathrm{e}}^{z/2}\mathrm{\Gamma }\left(1+a-b\right)}{\mathrm{\Gamma }\left(a\right)}\left(I_{b-1}\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{p_s(z)}{a^s}-\sqrt{z/a}{I}_b\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{q_s(z)}{a^s}\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.8.E12

Encodings:

pMML, png, TeX

Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added.

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

►

13.8.13 $$𝐌(-a,b,z)\sim {\left(z/a\right)}^{(1-b)/2}\frac{{\mathrm{e}}^{z/2}\mathrm{\Gamma }\left(1+a\right)}{\mathrm{\Gamma }\left(a+b\right)}\left(J_{b-1}\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{p_s(z)}{{(-a)}^s}-\sqrt{z/a}{J}_b\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{q_s(z)}{{(-a)}^s}\right),$$

ⓘ

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, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.8.13)

Permalink:

http://dlmf.nist.gov/13.8.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added. Expansion (13.8.13) is (10.3.58) in Temme (2015). In that reference the ${(-1)}^s$ is missing.

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

…

27: 10.43 Integrals

… ►

10.43.26 $${\int }_0^{\mathrm{\infty }}\frac{K_{\mu }\left(at\right)J_{\nu }\left(bt\right)}{t^{\lambda }}dt=\frac{b^{\nu }\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\lambda +\frac{1}{2}\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\lambda -\frac{1}{2}\mu +\frac{1}{2}\right)}{2^{\lambda +1}{a}^{\nu -\lambda +1}}𝐅(\frac{\nu -\lambda +\mu +1}{2},\frac{\nu -\lambda -\mu +1}{2};\nu +1;-\frac{b^2}{a^2}),$$ $\mathrm{\Re }\left(\nu +1-\lambda \right)>|\mathrm{\Re }\mu |,\mathrm{\Re }a>|\mathrm{\Im }b|$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{\Im }$: imaginary part, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/10.43.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

…

28: 13.16 Integral Representations

… ►

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

ⓘ

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

…

29: 18.12 Generating Functions

… ►

18.12.2 $${}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\alpha +1};\frac{(x-1)z}{2}){}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\beta +1};\frac{(x+1)z}{2})={\left(\frac{1}{2}(1-x)z\right)}^{-\frac{1}{2}\alpha }{J}_{\alpha }\left(\sqrt{2(1-x)z}\right){\left(\frac{1}{2}(1+x)z\right)}^{-\frac{1}{2}\beta }{I}_{\beta }\left(\sqrt{2(1+x)z}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{z}^n,$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (9.8.12))

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) for Equation (18.12.2)

Permalink:

http://dlmf.nist.gov/18.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

…

30: Bibliography O

… ►

Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms. SIAM J. Math. Anal. 22 (5), pp. 1475–1489.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (Hans-Jürgen Glaeske), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §13.7(iii), (9.7.18), (9.7.19), (9.7.20), (9.7.21), (9.7.22), (9.7.23), §9.7(v).

Encodings:

BibTeX

►

Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function. SIAM J. Math. Anal. 24 (3), pp. 756–767.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (Hussain S. Nur), ZentralBlatt

Referenced by:

§10.17(v), §13.7(iii), (9.7.18), (9.7.19), (9.7.20), (9.7.21), (9.7.22), (9.7.23), §9.7(v).

Encodings:

BibTeX

…