DLMF: Untitled Document

… ►In the double series the order of summation is important only when

j=1

. For further information on

\delta_{2j}

see §23.9: since the double sums in (20.6.6) and (23.9.1) are the same, we have

\delta_{2n}=c_{n}/(2n-1)

when

n\geq 2

.

… ►

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

… ►

M. Goano (1995) Algorithm 745: Computation of the complete and incomplete Fermi-Dirac integral. ACM Trans. Math. Software 21 (3), pp. 221–232.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 20D.
External Links:: ISSN 0098-3500, Document, GAMS (module 745; package TOMS), ZentralBlatt
Referenced by:: 6th item.
Encodings:: BibTeX

… ►

Z. Gong, L. Zejda, W. Dappen, and J. M. Aparicio (2001) Generalized Fermi-Dirac functions and derivatives: Properties and evaluation. Comput. Phys. Comm. 136 (3), pp. 294–309.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 20D
External Links:: Document, Code, ZentralBlatt
Referenced by:: 7th item.
Encodings:: BibTeX

…

… ►

M. R. Zaghloul (2016) Remark on “Algorithm 916: computing the Faddeyeva and Voigt functions”: efficiency improvements and Fortran translation. ACM Trans. Math. Softw. 42 (3), pp. 26:1–26:9.

ⓘ

Notes:: Includes MATLAB and Fortran programs claiming 6S or 13S accuracy for single and double precision
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry
Referenced by:: 3rd item, 6th item, §7.25(iii), §7.25(vi).
Encodings:: BibTeX

… ►

S. Zhang and J. Jin (1996) Computation of Special Functions. John Wiley & Sons Inc., New York.

ⓘ

Notes:: Includes diskette containing a large collection of mathematical function software written in Fortran. Implementation in double precision. Maximum accuracy 16S.
External Links:: ISBN 0-471-11963-6, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: 6th item, 3rd item, 1st item, 11st item, 7th item, 2nd item, 6th item, 3rd item, 2nd item, 5th item, 8th item, 4th item, 2nd item, §19.37(ii), §19.37(iii), §19.37(iii), §22.21, §24.19(i), §28.1, item (b), item (d), 8th item, 3rd item, 6th item, §5.22(ii), §5.22(iii), 2nd item, 2nd item, 4th item, 2nd item, 2nd item, §8.17(v), 4th item, 2nd item, 5th item, 3rd item, 3rd item, 3rd item, § ‣ 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, § ‣ 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, § ‣ 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, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

…

… ►

§22.6(ii) Double Argument

…

… ►Noble (2004) obtains double-precision accuracy for

W_{-\eta,\mu}\left(2\rho\right)

for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …

… ►

5.4.2

n!!=\begin{cases}2^{\frac{1}{2}n}\Gamma\left(\frac{1}{2}n+1\right),&n\text{ % even},\\ \pi^{-\frac{1}{2}}2^{\frac{1}{2}n+\frac{1}{2}}\Gamma\left(\frac{1}{2}n+1\right% ),&n\text{ odd}.\end{cases}

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $!!$ : double factorial and $n$ : nonnegative integer
Referenced by:: §5.4(i), §5.4(i)
Permalink:: http://dlmf.nist.gov/5.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §5.4(i), §5.4 and Ch.5

…

… ►

18.37.2

\iint\limits_{x^{2}+y^{2}<1}R^{(\alpha)}_{m,n}\left(x+\mathrm{i}y\right)R^{(% \alpha)}_{j,\ell}\left(x-\mathrm{i}y\right)\*(1-x^{2}-y^{2})^{\alpha}\,\mathrm% {d}x\,\mathrm{d}y=0,

m\neq j

and/or

n\neq\ell

.

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $R^{(\NVar{\alpha})}_{\NVar{m},\NVar{n}}\left(\NVar{z}\right)$ : disk polynomial, $\mathrm{i}$ : imaginary unit, $y$ : real variable, $\ell$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.37.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

… ►

18.37.4

\iint\limits_{x^{2}+y^{2}<1}R^{(\alpha)}_{m,n}\left(x+\mathrm{i}y\right)(x-iy)% ^{m-j}(x+iy)^{n-j}\*(1-x^{2}-y^{2})^{\alpha}\,\mathrm{d}x\,\mathrm{d}y=0,

j=1,2,\dots,\min(m,n)

;

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $R^{(\NVar{\alpha})}_{\NVar{m},\NVar{n}}\left(\NVar{z}\right)$ : disk polynomial, $\mathrm{i}$ : imaginary unit, $y$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.37.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

… ►

18.37.8

\iint\limits_{0<y<x<1}P^{\alpha,\beta,\gamma}_{m,n}\left(x,y\right)P^{\alpha,% \beta,\gamma}_{j,\ell}\left(x,y\right)\*(1-x)^{\alpha}(x-y)^{\beta}y^{\gamma}% \,\mathrm{d}x\,\mathrm{d}y=0,

m\neq j

and/or

n\neq\ell

.

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $P^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{m},\NVar{n}}\left(\NVar{x}% ,\NVar{y}\right)$ : triangle polynomial, $y$ : real variable, $\ell$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.37.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §18.37(ii), §18.37(ii), §18.37 and Ch.18

…

… ►

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

… ►

J. Segura and A. Gil (1998) Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments. Comput. Phys. Comm. 115 (1), pp. 69–86.

ⓘ

Notes:: Double-precision Fortran, minimum accuracy 14S, maximum accuracy 18D.
External Links:: ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt
Referenced by:: 3rd item.
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

… ►

W. V. Snyder (1993) Algorithm 723: Fresnel integrals. ACM Trans. Math. Software 19 (4), pp. 452–456.

ⓘ

Notes:: Single- and double-precision Fortran, maximum accuracy 16D. For remarks see same journal v. 22 (1996), p. 498-500 and v. 26 (2000), p. 617
External Links:: ISSN 0098-3500, Document, GAMS (module 723; package TOMS), ZentralBlatt
Referenced by:: 3rd item, 1st item.
Encodings:: BibTeX

…

… ►

•

Döring (1971) tabulates the first 100 values of $\nu$ $(>1)$ for which $J_{-\nu}'\left(x\right)$ has the double zero $x=\nu$ , 10D.

… ►

•

Kerimov and Skorokhodov (1985c) tabulates 201 double zeros of $J_{-\nu}''\left(x\right)$ , 10 double zeros of $J_{-\nu}'''\left(x\right)$ , 101 double zeros of $Y_{-\nu}'\left(x\right)$ , 201 double zeros of $Y_{-\nu}''\left(x\right)$ , and 10 double zeros of $Y_{-\nu}'''\left(x\right)$ , all to 8 or 9D.

… ►

•

Kerimov and Skorokhodov (1987) tabulates 100 complex double zeros $\nu$ of $Y_{\nu}'\left(ze^{-\pi i}\right)$ and ${H^{(1)}_{\nu}}'\left(ze^{-\pi i}\right)$ , 8D.

…

… ►

J. B. Campbell (1984) Determination of

\nu

-zeros of Hankel functions. Comput. Phys. Comm. 32 (3), pp. 333–339.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 7S.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

J. A. Christley and I. J. Thompson (1994) CRCWFN: Coupled real Coulomb wavefunctions. Comput. Phys. Comm. 79 (1), pp. 143–155.

ⓘ

Notes:: Double-precision Fortran code.
External Links:: ISSN 0010-4655, Document, Code, ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

… ►

D. S. Clemm (1969) Algorithm 352: Characteristic values and associated solutions of Mathieu’s differential equation. Comm. ACM 12 (7), pp. 399–407.

ⓘ

Notes:: Includes double-precision Fortran program, maximum accuracy 13D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §28.36(ii), §28.36(iii).
Encodings:: BibTeX

… ►

L. D. Cloutman (1989) Numerical evaluation of the Fermi-Dirac integrals. The Astrophysical Journal Supplement Series 71, pp. 677–699.

ⓘ

Notes:: Double-precision Fortran, maximum precision 12D.
External Links:: ISSN 0067-0049, Document, Code
Referenced by:: §25.12(iii), §25.18(i), 3rd item, 3rd item.
Encodings:: BibTeX

… ►

W. J. Cody (1983) Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Trans. Math. Software 9 (2), pp. 242–245.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 24S.
External Links:: ISSN 0098-3500, Document, GAMS (module 597; package TOMS), ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

…

double

31—40 of 93 matching pages

31: 20.6 Power Series

32: Bibliography G

33: Bibliography Z

34: 22.6 Elementary Identities

§22.6(ii) Double Argument

35: 33.23 Methods of Computation

36: 5.4 Special Values and Extrema

37: 18.37 Classical OP’s in Two or More Variables

38: Bibliography S

39: 10.75 Tables

40: Bibliography C