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21: Bibliography N

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Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes. ACM Trans. Math. Software 18 (3), pp. 345–349.

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Notes:

Double-precision Fortran, minimum accuracy 9S, maximum accuracy 13S.

External Links:

ISSN 0098-3500, Document, GAMS (module 707; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Notes:

Double precision. In Japanese, with some English.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

Encodings:

BibTeX

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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

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COULN, a program for evaluating negative energy Coulomb functions. Comput. Phys. Comm. 33 (4), pp. 413–419.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

2nd item, 7th item.

Encodings:

BibTeX

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

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Notes:

Double-precision Fortran-90 code.

External Links:

Document, Code

Referenced by:

§33.23(vi), 8th item.

Encodings:

BibTeX

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22: 1.6 Vectors and Vector-Valued Functions

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1.6.54 $${\iint }_{S}f(x,y,z)dS={\iint }_{D}f(𝚽(u,v))\Vert {𝐓}_u\times {𝐓}_v\Vert dudv.$$

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Symbols:

$dx$: differential of $x$ , $\Vert 𝐯\Vert$: vector norm ($l$²), $S$: parameterized surface, $𝚽(x,y,z)$: parameterization and $D$: open set in the plane

Permalink:

http://dlmf.nist.gov/1.6.E54

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6 and Ch.1

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1.6.55 $${\iint }_S𝐅\cdot d𝐒={\iint }_D𝐅\cdot ({𝐓}_u\times {𝐓}_v)dudv,$$

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Symbols:

$dx$: differential of $x$ , $S$: parameterized surface and $D$: open set in the plane

Permalink:

http://dlmf.nist.gov/1.6.E55

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6 and Ch.1

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1.6.56 $${\iint }_{{𝚽}_1(D_1)}𝐅\cdot d𝐒={\iint }_{{𝚽}_2(D_2)}𝐅\cdot d𝐒;$$

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Symbols:

$dx$: differential of $x$ , $𝚽(x,y,z)$: parameterization and $D$: open set in the plane

Permalink:

http://dlmf.nist.gov/1.6.E56

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6 and Ch.1

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1.6.57 $${\iint }_S(\nabla \times 𝐅)\cdot d𝐒={\int }_{\partial S}𝐅\cdot d𝐬,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\partial x$: partial differential of $x$ and $S$: parameterized surface

Permalink:

http://dlmf.nist.gov/1.6.E57

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6(v), §1.6 and Ch.1

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1.6.58 $${\iiint }_V(\nabla \cdot 𝐅)dV={\iint }_S𝐅\cdot d𝐒,$$

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Symbols:

$dx$: differential of $x$ , $S$: parameterized surface and $V$: closed region

Permalink:

http://dlmf.nist.gov/1.6.E58

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6(v), §1.6 and Ch.1

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23: 3.1 Arithmetics and Error Measures

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IEEE Standard

… ►In the case of the normalized binary interchange formats, the representation of data for binary32 (previously single precision) ($N=32$, $p=24$, $E_{\min }=-126$, $E_{\max }=127$), binary64 (previously double precision) ($N=64$, $p=53$, $E_{\min }=-1022$, $E_{\max }=1023$) and binary128 (previously quad precision) ($N=128$, $p=113$, $E_{\min }=-16382$, $E_{\max }=16383$) are as in Figure 3.1.1. … ►

Figure 3.1.1: Floating-point arithmetic. Representation of data in the binary interchange formats for binary32, binary64 and binary128 (previously single, double and quad precision).

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24: 20.5 Infinite Products and Related Results

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§20.5(iii) Double Products

… ►These double products are not absolutely convergent; hence the order of the limits is important. …

25: Bibliography Y

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Computation of Kummer functions $U(a,b,x)$ for large argument $x$ by using the $\tau$-method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).

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Notes:

Japanese with English summary. Double-precision Fortran.

External Links:

ISSN 0387-5806, FullText, MathReview

Referenced by:

§13.32(ii).

Encodings:

BibTeX

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26: 1.13 Differential Equations

… ►Here dots denote differentiations with respect to $\zeta$, and $\{z,\zeta \}$ is the Schwarzian derivative: ►

1.13.20 $$\{z,\zeta \}=-2{\dot{z}}^{1/2}\frac{d^2}{{d\zeta }^2}({\dot{z}}^{-1/2})=\frac{\stackrel{\dot{}\dot{}\dot{}}{z}}{\dot{z}}-\frac{3}{2}{\left(\frac{\ddot{z}}{\dot{z}}\right)}^2.$$

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Defines:

$\{z,\zeta \}$: Schwarzian derivative

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: variable, $\zeta (z)$: thrice-differentiable function, $\dot{z}$: derivative of $z$ with respect to $\zeta$ , $\ddot{z}$: 2nd derivative of $z$ with respect to $\zeta$ and $\stackrel{\dot{}\dot{}\dot{}}{z}$: 3rd derivative of $z$ with respect to $\zeta$

Permalink:

http://dlmf.nist.gov/1.13.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

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1.13.29 $$\ddot{w}(t)+\left(\lambda -\widehat{q}(t)\right)w(t)=0,$$ $t\in [0,c]$

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Symbols:

$[a,b]$: closed interval, $\in$: element of, $\widehat{q}(t)$: function, $w(t)$: solution, $\lambda$: real parameter, $t$: real variable, $c$: real variable and $\ddot{z}$: 2nd derivative of $w$ 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 $\frac{d^{2}w}{{dt}^2}$, via the transformation …

27: Bibliography F

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Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.

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Notes:

Single- and double-precision Fortran, maximum accuracy 32S.

External Links:

ISSN 0098-3500, Document, Code, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, 2nd item, 1st item.

Encodings:

BibTeX

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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An asymptotic expansion of the double gamma function. J. Approx. Theory 111 (2), pp. 298–314.

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External Links:

ISSN 0021-9045, Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§5.17.

Encodings:

BibTeX

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Computing the hypergeometric function. J. Comput. Phys. 137 (1), pp. 79–100.

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Notes:

Double-precision Fortran

External Links:

ISSN 0021-9991, Document, Code, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§15.19(i), 1st item, 2nd item.

Encodings:

BibTeX

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The PORT mathematical subroutine library. ACM Trans. Math. Software 4 (2), pp. 104–126.

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Notes:

Single- and double-precision Fortran.

External Links:

ISSN 0098-3500, Document

Referenced by:

§10.77(ii), §10.77(iv).

Encodings:

BibTeX

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28: Bibliography B

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A program for computing the Fermi-Dirac functions. Comput. Phys. Comm. 21 (3), pp. 315–322.

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Notes:

Double-precision Fortran, maximum accuracy 10D.

External Links:

Document, Code

Referenced by:

2nd item.

Encodings:

BibTeX

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

Encodings:

BibTeX

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The double confluent Heun equation: Characteristic exponent and connection formulae. Methods Appl. Anal. 1 (3), pp. 348–370.

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External Links:

ISSN 1073-2772, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§31.12.

Encodings:

BibTeX

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Expansions of Appell’s double hypergeometric functions. Quart. J. Math., Oxford Ser. 11, pp. 249–270.

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External Links:

Document, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§15.16, (16.15.4), §16.15, §16.16(i).

Encodings:

BibTeX

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Expansions of Appell’s double hypergeometric functions. II. Quart. J. Math., Oxford Ser. 12, pp. 112–128.

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External Links:

Document, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§16.16(i).

Encodings:

BibTeX

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29: 10.20 Uniform Asymptotic Expansions for Large Order

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§10.20(iii) Double Asymptotic Properties

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30: 16.15 Integral Representations and Integrals

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16.15.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)\mathrm{\Gamma }\left(\gamma -\beta -{\beta }^{\prime }\right)}{\iint }_{\mathrm{\Delta }}\frac{u^{\beta -1}{v}^{{\beta }^{\prime }-1}{(1-u-v)}^{\gamma -\beta -{\beta }^{\prime }-1}}{{(1-ux)}^{\alpha }{(1-vy)}^{{\alpha }^{\prime }}}dudv,$$ $\mathrm{\Re }\left(\gamma -\beta -{\beta }^{\prime }\right)>0$, $\mathrm{\Re }\beta >0$, $\mathrm{\Re }{\beta }^{\prime }>0$,

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Symbols:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Notes:

Erdélyi et al. (1953a, 5.8.1(3)) contains a typo.

Referenced by:

Erratum (V1.0.14) for Equation (16.15.3), Erratum (V1.0.14) for Equation (16.15.3)

Permalink:

http://dlmf.nist.gov/16.15.E3

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.14):

A stray “$f$” was removed; it appeared after the comma at the end of this equation.

See also:

Annotations for §16.15 and Ch.16

… ►For these and other formulas, including double Mellin–Barnes integrals, see Erdélyi et al. (1953a, §5.8). …