double integrals

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11: 2.4 Contour Integrals

… ►For double integrals with two coalescing stationary points see Qiu and Wong (2000). …

12: 31.9 Orthogonality

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31.9.2

\int_{\zeta}^{(1+,0+,1-,0-)}t^{\gamma-1}(1-t)^{\delta-1}(t-a)^{\epsilon-1}\*w_% {m}(t)w_{k}(t)\,\mathrm{d}t=\delta_{m,k}\theta_{m}.

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $\zeta$ : point and $\theta_{m}$ : normalization constant
Permalink:: http://dlmf.nist.gov/31.9.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.9(i), §31.9 and Ch.31

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13: 19.29 Reduction of General Elliptic Integrals

… ►Moreover, the requirement that one limit of integration be a branch point of the integrand is eliminated without doubling the number of standard integrals in the result. …

14: 5.17 Barnes’ $G$ -Function (Double Gamma Function)

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5.17.3

G\left(z+1\right)=(2\pi)^{z/2}\exp\left(-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\gamma z% ^{2}\right)\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp\left% (-z+\frac{z^{2}}{2k}\right)\right).

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Symbols:: $G\left(\NVar{z}\right)$ : Barnes’ $G$ -function (or double gamma function), $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $k$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.17.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §5.17 and Ch.5

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5.17.4

\operatorname{Ln}G\left(z+1\right)=\tfrac{1}{2}z\ln\left(2\pi\right)-\tfrac{1}% {2}z(z+1)+z\operatorname{Ln}\Gamma\left(z+1\right)-\int_{0}^{z}\operatorname{% Ln}\Gamma\left(t+1\right)\,\mathrm{d}t.

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Symbols:: $G\left(\NVar{z}\right)$ : Barnes’ $G$ -function (or double gamma function), $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.17.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §5.17 and Ch.5

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15: Bibliography H

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I. D. Hill (1973) Algorithm AS66: The normal integral. Appl. Statist. 22 (3), pp. 424–427.

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Notes:: Double-precision Fortran, maximum accuracy 12D.
External Links:: FullText, Code
Referenced by:: 5th item.
Encodings:: BibTeX

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16: Bibliography P

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R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:: Double-precision Fortran, maximum accuracy 20S.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 6th item.
Encodings:: BibTeX

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17: Bibliography S

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

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

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

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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

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I. A. Stegun and R. Zucker (1974) Automatic computing methods for special functions. II. The exponential integral

E_{n}(x)

. J. Res. Nat. Bur. Standards Sect. B 78B, pp. 199–216.

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Notes:: Double-precision Fortran, maximum accuracy 16S.
External Links:: MathReview (Bernard H. Rosman), ZentralBlatt
Referenced by:: §8.25(v), §8.28(vi).
Encodings:: BibTeX

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18: Bibliography

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D. E. Amos (1990) Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.

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Notes:: Double- and single-precision Fortran, maximum accuracy 18S or 14S.
External Links:: ISSN 0098-3500, Document, GAMS (module 683; package TOMS), MathReview, ZentralBlatt
Referenced by:: 1st item, 1st item.
Encodings:: BibTeX

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H. M. Antia (1993) Rational function approximations for Fermi-Dirac integrals. The Astrophysical Journal Supplement Series 84, pp. 101–108.

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Notes:: Double-precision Fortran, maximum precision 8D.
External Links:: ISSN 0067-0049, Document, Code
Referenced by:: 5th item, 1st item.
Encodings:: BibTeX

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19: 18.33 Polynomials Orthogonal on the Unit Circle

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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.

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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

20: Bibliography M

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K. L. Majumder and G. P. Bhattacharjee (1973) Algorithm AS 63. The incomplete beta integral. Appl. Statist. 22 (3), pp. 409–411.

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Notes:: Double-precision Fortran. For remark see same journal 26 (1977), pp. 111-114
External Links:: FullText, Code
Referenced by:: 4th item.
Encodings:: BibTeX

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