Mill ratio for complementary error function

SearchAdvancedHelp

(0.004 seconds)

31—40 of 963 matching pages

31: 14.1 Special Notation

§14.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise. ►The main functions treated in this chapter are the Legendre functions ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$; Ferrers functions ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$; conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ (also known as Mehler functions). …

32: 22.15 Inverse Functions

§22.15 Inverse Functions

►

§22.15(i) Definitions

… ►Each of these inverse functions is multivalued. The principal values satisfy … ►

33: 30.11 Radial Spheroidal Wave Functions

§30.11 Radial Spheroidal Wave Functions

►

§30.11(i) Definitions

… ►

Connection Formulas

… ►

§30.11(ii) Graphics

… ►

§30.11(iv) Wronskian

…

34: 35.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ► ►►

$a,b$	complex variables.
…

►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively ${\mathrm{\Gamma }}_m\left(a\right)$ and ${\mathrm{B}}_m(a,b)$, and the special functions of matrix argument: Bessel (of the first kind) $A_{\nu }\left(𝐓\right)$ and (of the second kind) $B_{\nu }\left(𝐓\right)$; confluent hypergeometric (of the first kind) ${}_1{}^{}F_1^{}(a;b;𝐓)$ or ${}_1{}^{}F_1^{}({\displaystyle \genfrac{}{}{0pt}{}{a}{b}};𝐓)$ and (of the second kind) $\mathrm{\Psi }(a;b;𝐓)$; Gaussian hypergeometric ${}_2{}^{}F_1^{}(a_1,a_2;b;𝐓)$ or ${}_2{}^{}F_1^{}({\displaystyle \genfrac{}{}{0pt}{}{a_1,a_2}{b}};𝐓)$; generalized hypergeometric ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}({\displaystyle \genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q}};𝐓)$. ►An alternative notation for the multivariate gamma function is ${\mathrm{\Pi }}_m(a)={\mathrm{\Gamma }}_m\left(a+\frac{1}{2}(m+1)\right)$ (Herz (1955, p. 480)). Related notations for the Bessel functions are ${𝒥}_{\nu +\frac{1}{2}(m+1)}(𝐓)=A_{\nu }\left(𝐓\right)/A_{\nu }\left(𝟎\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_m(0,\mathrm{\dots },0,\nu |𝐒,𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Terras (1988, pp. 49–64)), and ${𝒦}_{\nu }(𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Faraut and Korányi (1994, pp. 357–358)).

35: 22.2 Definitions

§22.2 Definitions

… ►where $K\left(k\right)$, $K^{\prime }\left(k\right)$ are defined in §19.2(ii). … ►As a function of $z$, with fixed $k$, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … … ►The Jacobian functions are related in the following way. …

36: 28.12 Definitions and Basic Properties

§28.12 Definitions and Basic Properties

… ►

§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

… ►If $q$ is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of $z$ and $q$ by the normalization … ► … ►

37: 1.16 Distributions

… ►

§1.16(i) Test Functions

… ► … ► … ► $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is called a distribution, or generalized function, if it is a continuous linear functional on $𝒟(I)$, that is, it is a linear functional and for every ${\varphi }_n\to \varphi$ in $𝒟(I)$, … ►

§1.16(iv) Heaviside Function

…

38: 16.17 Definition

§16.17 Definition

… ►Then the Meijer $G$-function is defined via the Mellin–Barnes integral representation: ►

16.17.1 $$G_{p,q}^{m,n}(z;𝐚;𝐛)=G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})=\frac{1}{2\pi \mathrm{i}}{\int }_L\left(\prod _{\mathrm{\ell }=1}^m\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)\prod _{\mathrm{\ell }=1}^n\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)/\left(\prod _{\mathrm{\ell }=m}^{q-1}\mathrm{\Gamma }\left(1-b_{\mathrm{\ell }+1}+s\right)\prod _{\mathrm{\ell }=n}^{p-1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }+1}-s\right)\right)\right)z^{s}ds,$$

ⓘ

Defines:

$G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $p$: 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:

Figure 16.17.1, Figure 16.17.1, §16.19

Permalink:

http://dlmf.nist.gov/16.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.17 and Ch.16

… ►When more than one of Cases (i), (ii), and (iii) is applicable the same value is obtained for the Meijer $G$-function. … ►Then …

39: 35.5 Bessel Functions of Matrix Argument

§35.5 Bessel Functions of Matrix Argument

►

§35.5(i) Definitions

… ►

§35.5(ii) Properties

… ►

§35.5(iii) Asymptotic Approximations

►For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).

40: 21.2 Definitions

… ►

§21.2(i) Riemann Theta Functions

… ► $\theta \left(𝐳\right|𝛀)$ is also referred to as a theta function with $g$ components, a $g$-dimensional theta function or as a genus $g$ theta function. … ►

§21.2(ii) Riemann Theta Functions with Characteristics

… ►This function is referred to as a Riemann theta function with characteristics $\left[\begin{array}{c}𝜶\\ 𝜷\end{array}\right]$. … ►

§21.2(iii) Relation to Classical Theta Functions

…