normalizing%20factor

(0.002 seconds)

1—10 of 258 matching pages

1: William P. Reinhardt

… ►Reinhardt firmly believes that the Mandelbrot set is a special function, and notes with interest that the natural boundaries of analyticity of many “more normal” special functions are also fractals. … ►

20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

2: 33.13 Complex Variable and Parameters

… ►

33.13.1

C_{\ell}\left(\eta\right)=2^{\ell}e^{\mathrm{i}{\sigma_{\ell}}\left(\eta\right% )-(\pi\eta/2)}\Gamma\left(\ell+1-\mathrm{i}\eta\right)/\Gamma\left(2\ell+2% \right),

ⓘ

Symbols:: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, ${\sigma_{\NVar{\ell}}}\left(\NVar{\eta}\right)$ : Coulomb phase shift, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ell$ : 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_{\ell}=(2\ell+1)C_{\ell}\left(\eta\right)/C_{\ell-1}\left(\eta\right).

ⓘ

Symbols:: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $\ell$ : nonnegative integer, $\eta$ : real parameter and $R_{\ell}$ : factor
Permalink:: http://dlmf.nist.gov/33.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §33.13 and Ch.33

…

3: 24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. In Wagstaff (2002) these results are extended to

n=60(2)152

and

n=40(2)88

, respectively, with further complete and partial factorizations listed up to

n=300

and

n=200

, respectively. …

4: Bibliography

… ►

A. G. Adams (1969) Algorithm 39: Areas under the normal curve. The Computer Journal 12 (2), pp. 197–198.

ⓘ

Notes:: Includes Algol program, maximum accuracy 12D.
External Links:: Document
Referenced by:: §7.25(ii).
Encodings:: BibTeX

… ►

A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

ⓘ

External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

… ►

V. I. Arnol’d (1972) Normal forms of functions near degenerate critical points, the Weyl groups

{A}_{k},{D}_{k},{E}_{k}

and Lagrangian singularities. Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).

ⓘ

Notes:: In Russian. English translation: Functional Anal. Appl., 6(1973), pp. 254–272.
External Links:: MathReview (G. R. Belickii), ZentralBlatt
Referenced by:: §36.2(i).
Encodings:: BibTeX

►

V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).

ⓘ

Notes:: Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. In Russian. English translation: Russian Math. Surveys, 29(1974), no. 2, pp. 10–50.
External Links:: Document, MathReview (G. R. Belickii), ZentralBlatt
Referenced by:: §36.2(i).
Encodings:: BibTeX

►

V. I. Arnol’d (1975) Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).

ⓘ

Notes:: In Russian. English translation: Russian Math. Surveys, 30(1975), no. 5, pp. 1–75.
External Links:: Document, MathReview (D. Siersma), ZentralBlatt
Referenced by:: §36.2(i), Ch.36.
Encodings:: BibTeX

…

5: 28.5 Second Solutions $\operatorname{fe}_{n}$ , $\operatorname{ge}_{n}$

… ►

28.5.1

\operatorname{fe}_{n}\left(z,q\right)=C_{n}(q)\left(z\operatorname{ce}_{n}% \left(z,q\right)+f_{n}(z,q)\right),

ⓘ

Defines:: $\operatorname{fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : second solution, Mathieu’s equation
Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $q=h^{2}$ : parameter, $n$ : integer, $z$ : complex variable, $f_{n}(z,q)$ : functions and $C_{n}(q)$ : factor
A&S Ref:: 20.3.6 (in different form)
Referenced by:: §28.5(i), §28.5(i)
Permalink:: http://dlmf.nist.gov/28.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

… ►The factors

C_{n}(q)

and

S_{n}(q)

in (28.5.1) and (28.5.2) are normalized so that ►

28.5.5

(C_{n}(q))^{2}\int_{0}^{2\pi}(f_{n}(x,q))^{2}\,\mathrm{d}x=(S_{n}(q))^{2}\int_% {0}^{2\pi}(g_{n}(x,q))^{2}\,\mathrm{d}x=\pi.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $q=h^{2}$ : parameter, $n$ : integer, $x$ : real variable, $f_{n}(z,q)$ : functions, $g_{n}(z,q)$ : functions, $C_{n}(q)$ : factor and $S_{n}(q)$ : factor
Permalink:: http://dlmf.nist.gov/28.5.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

… ►(Other normalizations for

C_{n}(q)

and

S_{n}(q)

can be found in the literature, but most formulas—including connection formulas—are unaffected since

\operatorname{fe}_{n}\left(z,q\right)/C_{n}(q)

and

\operatorname{ge}_{n}\left(z,q\right)/S_{n}(q)

are invariant.) … ►As a consequence of the factor

z

on the right-hand sides of (28.5.1), (28.5.2), all solutions of Mathieu’s equation that are linearly independent of the periodic solutions are unbounded as

z\to\pm\infty

\mathbb{R}

. …

6: 18.39 Applications in the Physical Sciences

… ►All are written in the same form as the product of three factors: the square root of a weight function

w(x)

, the corresponding OP or EOP, and constant factors ensuring unit normalization. … ►By Table 18.3.1#12 the normalized stationary states and corresponding eigenvalues are …With the normalization factor

\left(c\,h_{n}\right)^{-\ifrac{1}{2}}

the

\psi_{n}

are orthonormal in

L^{2}\left(\mathbb{R},\,\mathrm{d}x\right)

. … ►There is no need for a normalization constant here, as appropriate constants already appear in §18.36(vi). … ►Explicit normalization is given for the second, third, and fourth of these, paragraphs c) and d), below. …

7: 28.12 Definitions and Basic Properties

… ►In consequence, for the Floquet solutions

w(z)

the factor

e^{\pi\mathrm{i}\nu}

in (28.2.14) is no longer

\pm 1

. … ►

28.12.2

\lambda_{\nu}\left(-q\right)=\lambda_{\nu}\left(q\right)=\lambda_{-\nu}\left(q% \right).

ⓘ

Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §28.12(i), §28.12 and Ch.28

►As in §28.7 values of

q

for which (28.2.16) has simple roots

\lambda

are called normal values with respect to

\nu

. For real values of

\nu

and

q

all the

\lambda_{\nu}\left(q\right)

are real, and

q

is normal. … ►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 …

8: 3.6 Linear Difference Equations

… ►It therefore remains to apply a normalizing factor

\Lambda

. The process is then repeated with a higher value of

N

, and the normalized solutions compared. … ►The normalizing factor

\Lambda

can be the true value of

w_{0}

divided by its trial value, or

\Lambda

can be chosen to satisfy a known property of the wanted solution of the form … … ►For further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6). …

9: Bibliography B

… ►

G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

ⓘ

External Links:: Document
Referenced by:: §33.22(iv).
Encodings:: BibTeX

… ►

P. A. Becker (1997) Normalization integrals of orthogonal Heun functions. J. Math. Phys. 38 (7), pp. 3692–3699.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §31.9(i).
Encodings:: BibTeX

… ►

S. L. Belousov (1962) Tables of Normalized Associated Legendre Polynomials. Pergamon Press, The Macmillan Co., Oxford-New York.

ⓘ

External Links:: MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

… ►

E. Berti and V. Cardoso (2006) Quasinormal ringing of Kerr black holes: The excitation factors. Phys. Rev. D 74 (104020), pp. 1–27.

ⓘ

External Links:: ISSN 0556-2821, Document, MathReview
Referenced by:: 6th item.
Encodings:: BibTeX

… ►

D. M. Bressoud (1989) Factorization and Primality Testing. Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-97040-1, MathReview (Brigitte Vallée), ZentralBlatt
Referenced by:: §23.20(iii), §27.19, Profile David M. Bressoud.
Encodings:: BibTeX

…

10: Bibliography K

… ►

R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

ⓘ

External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

ⓘ

External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

… ►

N. Kimura (1988) On the degree of an irreducible factor of the Bernoulli polynomials. Acta Arith. 50 (3), pp. 243–249.

ⓘ

External Links:: ISSN 0065-1036, Pdf, MathReview (Wells Johnson), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

… ►

A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

ⓘ

External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

… ►

S. Koizumi (1976) Theta relations and projective normality of Abelian varieties. Amer. J. Math. 98 (4), pp. 865–889.

ⓘ

External Links:: FullText, MathReview (T. Oda), ZentralBlatt
Referenced by:: §21.6(ii).
Encodings:: BibTeX

…