DLMF: Untitled Document

… ►

L. Z. Salchev and V. B. Popov (1976) A property of the zeros of cross-product Bessel functions of different orders. Z. Angew. Math. Mech. 56 (2), pp. 120–121.

ⓘ

External Links:: Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §10.21(x).
Encodings:: BibTeX

… ►

M. J. Seaton (1982) Coulomb functions analytic in the energy. Comput. Phys. Comm. 25 (1), pp. 87–95.

ⓘ

Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: §33.1, §33.14(iv), 9th item.
Encodings:: BibTeX

… ►

N. T. Shawagfeh (1992) The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

ⓘ

External Links:: ISSN 0253-424X, MathReview
Referenced by:: §9.10(v).
Encodings:: BibTeX

►

B. L. Shea (1988) Algorithm AS 239. Chi-squared and incomplete gamma integral. Appl. Statist. 37 (3), pp. 466–473.

ⓘ

Notes:: Double-precision Fortran.
External Links:: FullText, Code
Referenced by:: 5th item.
Encodings:: BibTeX

… ►

P. N. Shivakumar and J. Xue (1999) On the double points of a Mathieu equation. J. Comput. Appl. Math. 107 (1), pp. 111–125.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §28.7.
Encodings:: BibTeX

…

… ►

18.33.31

\prod_{j=0}^{\infty}\left(1-\left|\alpha_{j}\right|^{2}\right)=\exp\left(\frac% {1}{2\pi\mathrm{i}}\int_{|z|=1}\ln\left(w(z)\right)\frac{\,\mathrm{d}z}{z}% \right).

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $w(x)$ : weight function and $z$ : complex variable
Proved:: Simon (2005a, (1.1.18))(proved)
Referenced by:: §18.33(vi)
Permalink:: http://dlmf.nist.gov/18.33.E31
Encodings:: pMML, png, TeX
See also:: Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

►By (18.33.25)

|\alpha_{j}|<1

, so the infinite product in (18.33.31) converges, although the limit may be zero. … ►

18.33.32

\sum_{j=0}^{\infty}|\alpha_{j}|^{2}<\infty\quad\Longleftrightarrow\quad\frac{1% }{2\pi\mathrm{i}}\int_{|z|=1}\ln\left((w(z)\right)\frac{\,\mathrm{d}z}{z}>-\infty.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $w(x)$ : weight function and $z$ : complex variable
Source:: Simon (2005a, (1.1.19))
Referenced by:: Erratum (V1.2.0) Section 18.33
Permalink:: http://dlmf.nist.gov/18.33.E32
Encodings:: pMML, png, TeX
See also:: Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

… ►

P. L. Walker (2012) Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.

ⓘ

External Links:: Document
Referenced by:: (20.7.34), §20.7(ix).
Encodings:: BibTeX

… ►

T. Watanabe, M. Natori, and T. Oguni (Eds.) (1994) Mathematical Software for the P.C. and Work Stations – A Collection of Fortran 77 Programs. North-Holland Publishing Co., Amsterdam.

ⓘ

Notes:: Translated from the 1989 Japanese original. With 1 IBM-PC floppy disk (5.25 inch) containing a large general purpose mathematical software collection written in Fortran, including some special function codes. Double precision.
External Links:: ISBN 0-444-82000-0, MathReview, ZentralBlatt
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

T. Weider (1999) Algorithm 794: Numerical Hankel transform by the Fortran program HANKEL. ACM Trans. Math. Software 25 (2), pp. 240–250.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 10S.
External Links:: ISSN 0098-3500, Document, GAMS (module 794; package TOMS), ZentralBlatt
Referenced by:: 7th item, 10th item.
Encodings:: BibTeX

… ►

J. Wishart (1928) The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A, pp. 32–52.

ⓘ

External Links:: FullText, Document, ZentralBlatt
Referenced by:: §35.3(i), §35.3(ii).
Encodings:: BibTeX

… ►

G. Wolf (1998) On the central connection problem for the double confluent Heun equation. Math. Nachr. 195, pp. 267–276.

ⓘ

External Links:: ISSN 0025-584X, MathReview, ZentralBlatt
Referenced by:: §31.12.
Encodings:: BibTeX

…

… ►

K

always has the form

T\times{\mathbb{Z}}^{r}

(Mordell’s Theorem: Silverman and Tate (1992, Chapter 3, §5)); the determination of

r

, the rank of

K

, raises questions of great difficulty, many of which are still open. …

T

must have one of the forms

\mathbb{Z}/(n\mathbb{Z})

,

1\leq n\leq 10

or

n=12

, or

(\mathbb{Z}/(2\mathbb{Z}))\times(\mathbb{Z}/(2n\mathbb{Z}))

,

1\leq n\leq 4

. …Given

P

, calculate

2P

,

4P

,

8P

by doubling as above. …

… ►

A. Gil and J. Segura (1997) Evaluation of Legendre functions of argument greater than one. Comput. Phys. Comm. 105 (2-3), pp. 273–283.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 18S
External Links:: ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

►

A. Gil and J. Segura (1998) A code to evaluate prolate and oblate spheroidal harmonics. Comput. Phys. Comm. 108 (2-3), pp. 267–278.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 18S
External Links:: Document, Code, ZentralBlatt
Referenced by:: 5th item, 1st item.
Encodings:: BibTeX

… ►

E. S. Ginsberg and D. Zaborowski (1975) Algorithm 490: The Dilogarithm function of a real argument [S22]. Comm. ACM 18 (4), pp. 200–202.

ⓘ

Notes:: Double-precision Fortran, minimum accuracy 15D. For remark see ACM Trans. Math. Software 2 (1976), p. 112
External Links:: ISSN 0001-0782, Document, Code
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

E. T. Goodwin (1949a) Recurrence relations for cross-products of Bessel functions. Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.

ⓘ

External Links:: ISSN 0033-5614, Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §10.6(iii).
Encodings:: BibTeX

… ►

H. P. W. Gottlieb (1985) On the exceptional zeros of cross-products of derivatives of spherical Bessel functions. Z. Angew. Math. Phys. 36 (3), pp. 491–494.

ⓘ

External Links:: ISSN 0044-2275, Document, MathReview (B. R. Bhonsle), ZentralBlatt
Referenced by:: §10.58.
Encodings:: BibTeX

…

… ►

J. Chen (1966) On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.

ⓘ

External Links:: MathReview (T. Hayashida)
Referenced by:: §27.13(ii).
Encodings:: BibTeX

… ►

J. A. Cochran (1964) Remarks on the zeros of cross-product Bessel functions. J. Soc. Indust. Appl. Math. 12 (3), pp. 580–587.

ⓘ

External Links:: Document, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §10.21(x), §10.21(x).
Encodings:: BibTeX

… ►

J. A. Cochran (1966a) The analyticity of cross-product Bessel function zeros. Proc. Cambridge Philos. Soc. 62, pp. 215–226.

ⓘ

External Links:: MathReview (R. G. Langebartel), ZentralBlatt
Referenced by:: §10.21(x).
Encodings:: BibTeX

►

J. A. Cochran (1966b) The asymptotic nature of zeros of cross-product Bessel functions. Quart. J. Mech. Appl. Math. 19 (4), pp. 511–522.

ⓘ

External Links:: ISSN 0033-5614, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §10.21(x).
Encodings:: BibTeX

… ►

W. C. Connett, C. Markett, and A. L. Schwartz (1993) Product formulas and convolutions for angular and radial spheroidal wave functions. Trans. Amer. Math. Soc. 338 (2), pp. 695–710.

ⓘ

External Links:: ISSN 0002-9947, FullText, MathReview (Jean-Claude Lucquiaud), ZentralBlatt
Referenced by:: §30.10, §30.11(vi).
Encodings:: BibTeX

…

… ►Here dots denote differentiations with respect to

\zeta

, and

\left\{z,\zeta\right\}

is the Schwarzian derivative: … ►

§1.13(v) Products of Solutions

►The product of any two solutions of (1.13.1) satisfies … ►

1.13.29

\ddot{w}(t)+\left(\lambda-\widehat{q}(t)\right)w(t)=0,

t\in[0,c]

ⓘ

Symbols:: $[\NVar{a},\NVar{b}]$ : closed interval, $\in$ : element of, $\widehat{q}(t)$ : function, $w(t)$ : solution, $\lambda$ : real parameter, $t$ : real variable, $c$ : real variable and $\ddot{\NVar{z}}$ : 2nd derivative of $\NVar{w}$ with respect to $t$
Referenced by:: §1.13(viii), §1.18
Permalink:: http://dlmf.nist.gov/1.13.E29
Encodings:: pMML, png, TeX
See also:: Annotations for §1.13(viii), §1.13(viii), §1.13 and Ch.1

►where

\ddot{w}

now denotes

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}

, via the transformation …

… ►

NetNUMPAC (free Fortran library)

ⓘ

Notes:: Double precision. In Japanese, with some English.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

M. M. Nieto and L. M. Simmons (1979) Eigenstates, coherent states, and uncertainty products for the Morse oscillator. Phys. Rev. A (3) 19 (2), pp. 438–444.

ⓘ

External Links:: ISSN 1050-2947, Document, Link, MathReview Entry
Referenced by:: (18.39.17), (18.39.18).
Encodings:: BibTeX

… ►

NMS (free collection of Fortran subroutines)

ⓘ

Notes:: Single and double precision. See Kahaner et al. (1989).
External Links:: GAMS (package NMS)
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

►

C. J. Noble and I. J. Thompson (1984) COULN, a program for evaluating negative energy Coulomb functions. Comput. Phys. Comm. 33 (4), pp. 413–419.

ⓘ

Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: 2nd item, 7th item.
Encodings:: BibTeX

►

C. J. Noble (2004) Evaluation of negative energy Coulomb (Whittaker) functions. Comput. Phys. Comm. 159 (1), pp. 55–62.

ⓘ

Notes:: Double-precision Fortran-90 code.
External Links:: Document, Code
Referenced by:: §33.23(vi), 8th item.
Encodings:: BibTeX

…

… ►The determinant of an upper or lower triangular, or diagonal, square matrix

\mathbf{A}

is the product of the diagonal elements

\det(\mathbf{A})=\prod_{i=1}^{n}a_{ii}

. … ►

1.3.14

\det\left[\frac{1}{a_{j}-b_{k}}\right]=(-1)^{n(n-1)/2}\*\prod_{1\leq j<k\leq n% }(a_{k}-a_{j})(b_{k}-b_{j})\Bigg{/}\prod^{n}_{j,k=1}(a_{j}-b_{k}).

ⓘ

Symbols:: $\det$ : determinant, $j$ : integer, $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

… ►These have the property that the double series … ►The adjoint of a matrix

\mathbf{A}

is the matrix

{\mathbf{A}}^{*}

such that

\left\langle\mathbf{A}\mathbf{a},\mathbf{b}\right\rangle=\left\langle\mathbf{a% },{\mathbf{A}}^{*}\mathbf{b}\right\rangle

for all

\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}

. … ►

1.3.20

\mathbf{u}=\sum_{i=1}^{n}c_{i}\mathbf{a}_{i},

c_{i}=\left\langle\mathbf{u},\mathbf{a}_{i}\right\rangle

.

ⓘ

Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors and $n$ : nonnegative integer
Referenced by:: Erratum (V1.2.0) Section 1.3
Permalink:: http://dlmf.nist.gov/1.3.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(iv), §1.3(iv), §1.3 and Ch.1

…

… ►and the coefficients

\mathsf{A}_{s}(\tau)

are the product of

\tau^{s}

and a polynomial in

\tau

of degree

2s

. … ►

\mathsf{A}_{2}(\tau)=\tfrac{1}{288}\tau^{2}(6160\tau^{4}+18480\tau^{3}+19404% \tau^{2}+8028\tau+945).

… ►In addition, it enjoys a double asymptotic property: it holds if either or both

\mu

and

t

tend to infinity. …The proof of the double asymptotic property then follows with the aid of error bounds; compare §10.41(iv). …

double products

11—20 of 30 matching pages

11: Bibliography S

12: 18.33 Polynomials Orthogonal on the Unit Circle

13: Bibliography W

14: 23.20 Mathematical Applications

15: Bibliography G

16: Bibliography C

17: 1.13 Differential Equations

§1.13(v) Products of Solutions

18: Bibliography N

19: 1.3 Determinants, Linear Operators, and Spectral Expansions

20: 12.10 Uniform Asymptotic Expansions for Large Parameter