hypergeometric functions

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41—50 of 222 matching pages

44: 35.1 Special Notation

$a, b$	complex variables.
…

►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively

\Gamma_{m}\left(a\right)

and

\mathrm{B}_{m}\left(a,b\right)

, and the special functions of matrix argument: Bessel (of the first kind)

A_{\nu}\left(\mathbf{T}\right)

and (of the second kind)

B_{\nu}\left(\mathbf{T}\right)

; confluent hypergeometric (of the first kind)

{{}_{1}F_{1}}\left(a;b;\mathbf{T}\right)

\displaystyle{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)

and (of the second kind)

\Psi\left(a;b;\mathbf{T}\right)

; Gaussian hypergeometric

{{}_{2}F_{1}}\left(a_{1},a_{2};b;\mathbf{T}\right)

\displaystyle{{}_{2}F_{1}}\left({a_{1},a_{2}\atop b};\mathbf{T}\right)

; generalized hypergeometric

{{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)

\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% \mathbf{T}\right)

. … ►Related notations for the Bessel functions are

\mathcal{J}_{\nu+\frac{1}{2}(m+1)}(\mathbf{T})=A_{\nu}\left(\mathbf{T}\right)/% A_{\nu}\left(\boldsymbol{{0}}\right)

(Faraut and Korányi (1994, pp. 320–329)),

K_{m}(0,\dots,0,\nu\mathpunct{|}\mathbf{S},\mathbf{T})=\left|\mathbf{T}\right|% ^{\nu}B_{\nu}\left(\mathbf{S}\mathbf{T}\right)

(Terras (1988, pp. 49–64)), and

\mathcal{K}_{\nu}(\mathbf{T})=\left|\mathbf{T}\right|^{\nu}B_{\nu}\left(% \mathbf{S}\mathbf{T}\right)

(Faraut and Korányi (1994, pp. 357–358)).

45: 7.11 Relations to Other Functions

… ►

Confluent Hypergeometric Functions

… ►

7.11.4

\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}M\left(\tfrac{1}{2},\tfrac{3}{2},-z^{% 2}\right)=\frac{2z}{\sqrt{\pi}}e^{-z^{2}}M\left(1,\tfrac{3}{2},z^{2}\right),

ⓘ

Symbols:: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable
A&S Ref:: 7.1.21
Permalink:: http://dlmf.nist.gov/7.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §7.11, §7.11 and Ch.7

►

7.11.5

\operatorname{erfc}z=\frac{1}{\sqrt{\pi}}e^{-z^{2}}U\left(\tfrac{1}{2},\tfrac{% 1}{2},z^{2}\right)=\frac{z}{\sqrt{\pi}}e^{-z^{2}}U\left(1,\tfrac{3}{2},z^{2}% \right).

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable
Referenced by:: §7.11
Permalink:: http://dlmf.nist.gov/7.11.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §7.11, §7.11 and Ch.7

►

7.11.6

C\left(z\right)+iS\left(z\right)=zM\left(\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2% }\pi iz^{2}\right)=ze^{\pi iz^{2}/2}M\left(1,\tfrac{3}{2},-\tfrac{1}{2}\pi iz^% {2}\right).

ⓘ

Symbols:: $C\left(\NVar{z}\right)$ : Fresnel integral, $S\left(\NVar{z}\right)$ : Fresnel integral, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/7.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §7.11, §7.11 and Ch.7

►

Generalized Hypergeometric Functions

…

46: 10.16 Relations to Other Functions

… ►

Confluent Hypergeometric Functions

►

10.16.5

J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}e^{\mp iz}}{\Gamma\left(\nu+1% \right)}M\left(\nu+\tfrac{1}{2},2\nu+1,\pm 2iz\right),

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.69 (modified)
Referenced by:: §10.16, §10.39
Permalink:: http://dlmf.nist.gov/10.16.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §10.16, §10.16 and Ch.10

… ►For the functions

M

and

U

see §13.2(i). …For the functions

M_{0,\nu}

and

W_{0,\nu}

see §13.14(i). … ►

Generalized Hypergeometric Functions

…

47: 15.19 Methods of Computation

§15.19 Methods of Computation

… ►For

z\in\mathbb{R}

it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval

[0,\frac{1}{2}]

. … ►The representation (15.6.1) can be used to compute the hypergeometric function in the sector

|\operatorname{ph}\left(1-z\right)|<\pi

. … ► … ►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of the Gauss hypergeometric function, when the variable and parameters are real and one of the numerator parameters is a positive integer. …

48: 16.5 Integral Representations and Integrals

§16.5 Integral Representations and Integrals

… ►Lastly, when

p>q+1

the right-hand side of (16.5.1) can be regarded as the definition of the (customarily undefined) left-hand side. In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as

z\to 0

in the sector

|\operatorname{ph}\left(-z\right)|\leq(p+1-q-\delta)\pi/2

, where

\delta

is an arbitrary small positive constant. … ►Laplace transforms and inverse Laplace transforms of generalized hypergeometric functions are given in Prudnikov et al. (1992a, §3.38) and Prudnikov et al. (1992b, §3.36). …

49: 17.4 Basic Hypergeometric Functions

§17.4 Basic Hypergeometric Functions

… ►

§17.4(ii) ${{}_{r}\psi_{s}}$ Functions

… ►For the function

{{}_{r}H_{r}}

see §16.4(v). … ►The series (17.4.1) is said to be balanced or Saalschützian when it terminates,

r=s

z=q

, and … ►The series (17.4.1) is said to be k-balanced when

r=s

and …

50: 13.1 Special Notation

… ►The main functions treated in this chapter are the Kummer functions

M\left(a,b,z\right)

and

U\left(a,b,z\right)

, Olver’s function

{\mathbf{M}}\left(a,b,z\right)

, and the Whittaker functions

M_{\kappa,\mu}\left(z\right)

and

W_{\kappa,\mu}\left(z\right)

. … ►

hypergeometric functions

41—50 of 222 matching pages

41: 15.4 Special Cases

§15.4(i) Elementary Functions

§15.4(ii) Argument Unity

Chu–Vandermonde Identity

§15.4(iii) Other Arguments

42: 16.9 Zeros

§16.9 Zeros

43: 12.20 Approximations

44: 35.1 Special Notation

45: 7.11 Relations to Other Functions

Confluent Hypergeometric Functions

Generalized Hypergeometric Functions

46: 10.16 Relations to Other Functions

Confluent Hypergeometric Functions

Generalized Hypergeometric Functions

47: 15.19 Methods of Computation

§15.19 Methods of Computation

48: 16.5 Integral Representations and Integrals

§16.5 Integral Representations and Integrals

49: 17.4 Basic Hypergeometric Functions

§17.4 Basic Hypergeometric Functions

§17.4(ii) ${{}_{r}\psi_{s}}$ Functions

50: 13.1 Special Notation

hypergeometric functions

41—50 of 222 matching pages

41: 15.4 Special Cases

§15.4(i) Elementary Functions

§15.4(ii) Argument Unity

Chu–Vandermonde Identity

§15.4(iii) Other Arguments

42: 16.9 Zeros

§16.9 Zeros

43: 12.20 Approximations

44: 35.1 Special Notation

45: 7.11 Relations to Other Functions

Confluent Hypergeometric Functions

Generalized Hypergeometric Functions

46: 10.16 Relations to Other Functions

Confluent Hypergeometric Functions

Generalized Hypergeometric Functions

47: 15.19 Methods of Computation

§15.19 Methods of Computation

48: 16.5 Integral Representations and Integrals

§16.5 Integral Representations and Integrals

49: 17.4 Basic Hypergeometric Functions

§17.4 Basic Hypergeometric Functions

§17.4(ii) ψ s r Functions

50: 13.1 Special Notation

§17.4(ii) ${{}_{r}\psi_{s}}$ Functions