normalization

SearchAdvancedHelp

(0.000 seconds)

1—10 of 108 matching pages

1: 29.21 Tables

… ►

Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^2=0.1(.1)0.9$, $n=1(1)30$. Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.

2: 29.20 Methods of Computation

… ►The normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5). … ►Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … ►§29.15(i) includes formulas for normalizing the eigenvectors. …

3: 35.9 Applications

… ►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument ${}_p{}^{}F_q^{}$, with $p\le 2$ and $q\le 1$. … ►For applications of the integral representation (35.5.3) see McFarland and Richards (2001, 2002) (statistical estimation of misclassification probabilities for discriminating between multivariate normal populations). …

4: 7.1 Special Notation

… ►The notations $P(z)$, $Q(z)$, and $\mathrm{\Phi }(z)$ are used in mathematical statistics, where these functions are called the normal or Gaussian probability functions.

5: 33.13 Complex Variable and Parameters

… ►

33.13.1 $$C_{\mathrm{\ell }}\left(\eta \right)=2^{\mathrm{\ell }}{\mathrm{e}}^{\mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)-(\pi \eta /2)}\mathrm{\Gamma }\left(\mathrm{\ell }+1-\mathrm{i}\eta \right)/\mathrm{\Gamma }\left(2\mathrm{\ell }+2\right),$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

… ►

33.13.2 $$R_{\mathrm{\ell }}=(2\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right)/C_{\mathrm{\ell }-1}\left(\eta \right).$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $R_{\mathrm{\ell }}$: factor

Permalink:

http://dlmf.nist.gov/33.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

…

6: 8.23 Statistical Applications

… ►The function ${\mathrm{B}}_x(a,b)$ and its normalization $I_x(a,b)$ play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). …

7: 23.23 Tables

… ►2 in Abramowitz and Stegun (1964) gives values of $\mathrm{\wp }\left(z\right)$, ${\mathrm{\wp }}^{\prime }\left(z\right)$, and $\zeta \left(z\right)$ to 7 or 8D in the rectangular and rhombic cases, normalized so that ${\omega }_1=1$ and ${\omega }_3=\mathrm{i}a$ (rectangular case), or ${\omega }_1=1$ and ${\omega }_3=\frac{1}{2}+\mathrm{i}a$ (rhombic case), for $a$ = 1. …

8: 28.7 Analytic Continuation of Eigenvalues

… ►The branch points are called the exceptional values, and the other points normal values. The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). All real values of $q$ are normal values. …

9: 8.2 Definitions and Basic Properties

… ►

8.2.3 $$\gamma (a,z)+\mathrm{\Gamma }(a,z)=\mathrm{\Gamma }\left(a\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.3

Referenced by:

§8.6(i), §8.7

Permalink:

http://dlmf.nist.gov/8.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(i), §8.2 and Ch.8

►Normalized functions are: … ►

8.2.5 $$P(a,z)+Q(a,z)=1.$$

ⓘ

Symbols:

$P(a,z)$: normalized incomplete gamma function, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(i), §8.2 and Ch.8

… ►

8.2.6 $${\gamma }^{\ast }(a,z)=z^{-a}P(a,z)=\frac{z^{-a}}{\mathrm{\Gamma }\left(a\right)}\gamma (a,z).$$

ⓘ

Defines:

${\gamma }^{\ast }(a,z)$: incomplete gamma function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma (a,z)$: incomplete gamma function, $P(a,z)$: normalized incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.4

Referenced by:

§8.2(iii), §8.6(i)

Permalink:

http://dlmf.nist.gov/8.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(i), §8.2 and Ch.8

…

10: 28.14 Fourier Series

… ►and the normalization relation ►

28.14.5 $$\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{\left(c_{2m}^{\nu }(q)\right)}^2=1;$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Referenced by:

item (d)

Permalink:

http://dlmf.nist.gov/28.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

…