Gaussian

SearchAdvancedHelp

(0.001 seconds)

11—20 of 34 matching pages

11: 20.13 Physical Applications

… ►Then the nonperiodic Gaussian …Thus the classical theta functions are “periodized”, or “anti-periodized”, Gaussians; see Bellman (1961, pp. 18, 19). …

12: 35.1 Special Notation

… ► ►►

$a,b$	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 ${}_p{}^{}F_q^{}({\displaystyle \genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q}};𝐓)$. … ►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)).

13: 7.21 Physical Applications

… ►Voigt functions $𝖴(x,t)$, $𝖵(x,t)$, can be regarded as the convolution of a Gaussian and a Lorentzian, and appear when the analysis of light (or particulate) absorption (or emission) involves thermal motion effects. …

14: 26.16 Multiset Permutations

… ►The $q$-multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by ►

26.16.1 $${\left[\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right]}_q=\prod _{k=1}^{n-1}{\left[\genfrac{}{}{0.0pt}{}{a_k+a_{k+1}+\mathrm{\cdots }+a_n}{a_k}\right]}_q,$$

ⓘ

Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), ${\left[\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right]}_q$: $q$-multinomial coefficient, $k$: nonnegative integer, $n$: nonnegative integer, $a_j$: numbers and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.16 and Ch.26

…

15: 17.2 Calculus

… ►

17.2.27 $${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q=\frac{{(q;q)}_n}{{(q;q)}_m{(q;q)}_{n-m}}=\frac{{(q^{-n};q)}_m{(-1)}^m{q}^{nm-\left(\genfrac{}{}{0.0pt}{}{m}{2}\right)}}{{(q;q)}_m},$$

ⓘ

Defines:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial)

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(ii), §17.2 and Ch.17

►

17.2.28 $$\underset{q\to 1}{lim}{\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q=\left(\genfrac{}{}{0.0pt}{}{n}{m}\right)=\frac{n!}{m!(n-m)!},$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(ii), §17.2 and Ch.17

►

17.2.29 $${\left[\genfrac{}{}{0.0pt}{}{m+n}{m}\right]}_q=\frac{{(q^{n+1};q)}_m}{{(q;q)}_m},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(ii), §17.2 and Ch.17

►

17.2.30 $${\left[\genfrac{}{}{0.0pt}{}{-n}{m}\right]}_q={\left[\genfrac{}{}{0.0pt}{}{m+n-1}{m}\right]}_q{(-1)}^m{q}^{-mn-\left(\genfrac{}{}{0.0pt}{}{m}{2}\right)},$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(ii), §17.2 and Ch.17

►

17.2.31 $${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q={\left[\genfrac{}{}{0.0pt}{}{n-1}{m-1}\right]}_q+q^m{\left[\genfrac{}{}{0.0pt}{}{n-1}{m}\right]}_q,$$

ⓘ

Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(ii), §17.2 and Ch.17

…

16: 9.17 Methods of Computation

… ►For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979). …

17: Philip J. Davis

… ►An interesting anecdote told by Davis reveals that he and mathematician Philip Rabinowitz were dubbed “Heroes of the SEAC” when their Gaussian integration code executed correctly on its first run. …

18: 17.3 $q$-Elementary and $q$-Special Functions

… ►

17.3.8 $$A_{m,s}\left(q\right)=q^{\left(\genfrac{}{}{0.0pt}{}{s-m}{2}\right)+\left(\genfrac{}{}{0.0pt}{}{s}{2}\right)}\sum _{j=0}^s{(-1)}^j{q}^{\left(\genfrac{}{}{0.0pt}{}{j}{2}\right)}{\left[\genfrac{}{}{0.0pt}{}{m+1}{j}\right]}_q\frac{{(1-q^{s-j})}^m}{{(1-q)}^m}.$$

ⓘ

Defines:

$A_{m,s}\left(q\right)$: $q$-Euler number

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer, $m$: nonnegative integer and $s$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(iii), §17.3(iii), §17.3 and Ch.17

… ►

17.3.9 $$a_{m,s}\left(q\right)=\frac{q^{-\left(\genfrac{}{}{0.0pt}{}{s}{2}\right)}{(1-q)}^s}{{(q;q)}_s}\sum _{j=0}^s{(-1)}^j{q}^{\left(\genfrac{}{}{0.0pt}{}{j}{2}\right)}{\left[\genfrac{}{}{0.0pt}{}{s}{j}\right]}_q\frac{{(1-q^{s-j})}^m}{{(1-q)}^m}.$$

ⓘ

Defines:

$a_{m,s}\left(q\right)$: $q$-Stirling number

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer, $m$: nonnegative integer and $s$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.3.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(iii), §17.3(iii), §17.3 and Ch.17

…

19: 26.1 Special Notation

… ► ►►►

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$	binomial coefficient.
…
${\left[\genfrac{}{}{0.0pt}{}{m}{n}\right]}_q$	Gaussian polynomial.
…

…

20: Bibliography F

… ►

The Plasma Dispersion Function: The Hilbert Transform of the Gaussian. Academic Press, London-New York.

ⓘ

Notes:

Erratum: Math. Comp. v. 26 (1972), no. 119, p. 814.

External Links:

MathReview (R. Balescu)

Referenced by:

§7.21.

Encodings:

BibTeX

… ►

Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond. In Recent Perspectives in Random Matrix Theory and Number Theory, London Math. Soc. Lecture Note Ser., Vol. 322, pp. 31–78.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§18.38(ii).

Encodings:

BibTeX