double integrals

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21: Bibliography C

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Notes:

This library treats about 180 different mathematical functions, including a substantial collection of probability integrals, Bessel functions, and higher transcendental functions. Implementations in single, double, and quad precisions are provided. See Moshier (1989).

External Links:

GAMS (package CEPHES)

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, § ‣ 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

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Numerical evaluation of the Fermi-Dirac integrals. The Astrophysical Journal Supplement Series 71, pp. 677–699.

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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

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22: Bibliography G

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

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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

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23: 31.10 Integral Equations and Representations

§31.10 Integral Equations and Representations

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Kernel Functions

… ►where $\mathrm{\Re }\gamma >0$, $\mathrm{\Re }\delta >0$, and $C$ be the Pochhammer double-loop contour about 0 and 1 (as in §31.9(i)). … ►

Kernel Functions

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24: 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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25: 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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26: 15.6 Integral Representations

§15.6 Integral Representations

►The function $𝐅(a,b;c;z)$ (not $F(a,b;c;z)$) has the following integral representations: … ►Note that (15.6.8) can be rewritten as a fractional integral. … ►

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Figure 15.6.1: $t$-plane. … Magnify

27: 1.14 Integral Transforms

§1.14 Integral Transforms

… ►where the last integral denotes the Cauchy principal value (1.4.25). … ►

Laplace Transform

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§1.14(viii) Compendia

►For more extensive tables of the integral transforms of this section and tables of other integral transforms, see Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000), Marichev (1983), Oberhettinger (1972, 1974, 1990), Oberhettinger and Badii (1973), Oberhettinger and Higgins (1961), Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

28: 1.13 Differential Equations

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1.13.16 $$\eta =\int \exp \left(-\int f(z)dz\right)dz.$$

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Defines:

$\eta$: change of variable (locally)

Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $z$: variable and $f(z)$: analytic coefficient

Permalink:

http://dlmf.nist.gov/1.13.E16

Encodings:

pMML, png, TeX

See also:

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

… ►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 …

29: Bibliography R

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A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

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Notes:

Double-precision Fortran, maximum accuracy 18D.

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

6th item.

Encodings:

BibTeX

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Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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External Links:

ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

Encodings:

BibTeX

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Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

Encodings:

BibTeX

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Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v).

Encodings:

BibTeX

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Dawson’s integral and the sampling theorem. Computers in Physics 3 (2), pp. 85–87.

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Referenced by:

§7.25(vi).

Encodings:

BibTeX

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30: Guide to Searching the DLMF

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Table 1: Query Examples

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Query	Matching records contain
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int sin	the integral of the sin function
int_$^$ sin	any definite integral of sin
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any double-quoted sequence of textual words and numbers.

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Table 2: Wildcard Examples

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Query	What it stands for
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int_$^$ sin	any definite integral of sin.

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Table 3: A sample of recognized symbols

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Symbols	Comments
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->, <-, <->, =>, <==, <=>	For arrows $\to$, $\leftarrow$, $\leftrightarrow$, $\Rightarrow$, $\Leftarrow$ and $\iff$
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