Laplace%20integral

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1: 20.10 Integrals

§20.10 Integrals

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§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

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20.10.4

\int_{0}^{\infty}e^{-st}\theta_{1}\left(\frac{\beta\pi}{2\ell}\middle|\frac{i% \pi t}{\ell^{2}}\right)\,\mathrm{d}t=\int_{0}^{\infty}e^{-st}\theta_{2}\left(% \frac{(1+\beta)\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\,\mathrm{d}t=% -\frac{\ell}{\sqrt{s}}\sinh\left(\beta\sqrt{s}\right)\operatorname{sech}\left(% \ell\sqrt{s}\right),

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $s$ : parameter, $\beta$ : parameter and $\ell$ : parameter
Permalink:: http://dlmf.nist.gov/20.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §20.10(ii), §20.10 and Ch.20

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20.10.5

\int_{0}^{\infty}e^{-st}\theta_{3}\left(\frac{(1+\beta)\pi}{2\ell}\middle|% \frac{i\pi t}{\ell^{2}}\right)\,\mathrm{d}t=\int_{0}^{\infty}e^{-st}\theta_{4}% \left(\frac{\beta\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\,\mathrm{d}% t=\frac{\ell}{\sqrt{s}}\cosh\left(\beta\sqrt{s}\right)\operatorname{csch}\left% (\ell\sqrt{s}\right).

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $s$ : parameter, $\beta$ : parameter and $\ell$ : parameter
Permalink:: http://dlmf.nist.gov/20.10.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §20.10(ii), §20.10 and Ch.20

… ►For further integrals of theta functions see Erdélyi et al. (1954a, pp. 61–62 and 339), Prudnikov et al. (1990, pp. 356–358), Prudnikov et al. (1992a, §3.41), and Gradshteyn and Ryzhik (2000, pp. 627–628).

2: Bibliography N

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D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:: ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §28.26(ii).
Encodings:: BibTeX

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W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: §19.37(iv).
Encodings:: BibTeX

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G. Nemes (2013a) An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.

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External Links:: ISSN 0176-4276, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt
Referenced by:: §5.11(i), §5.11(i).
Encodings:: BibTeX

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G. Nemes (2020) An extension of Laplace’s method. Constr. Approx. 51 (2), pp. 247–272.

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External Links:: ISSN 0176-4276, Document, Link, MathReview (José Luis López), ZentralBlatt
Referenced by:: (11.11.16), (11.11.17), §11.11(iii).
Encodings:: BibTeX

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E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:: See for additions and corrections same journal, Sect. B 75, 149–163 (1975)
External Links:: MathReview, ZentralBlatt
Referenced by:: §7.14(iii), §7.21, §7.7(iii).
Encodings:: BibTeX

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3: Bibliography S

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K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:: ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §22.19(i), §22.20(vi).
Encodings:: BibTeX

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H. E. Salzer (1955) Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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External Links:: FullText, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §3.5(viii).
Encodings:: BibTeX

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J. L. Schiff (1999) The Laplace Transform: Theory and Applications. Undergraduate Texts in Mathematics, Springer-Verlag, New York.

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External Links:: ISBN 0-387-98698-7, MathReview (Philip Heywood), ZentralBlatt
Referenced by:: §1.14(iii), §1.14(vii).
Encodings:: BibTeX

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A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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

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

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4: Bibliography D

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C. de la Vallée Poussin (1896a) Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction

\zeta(s)

de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

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Notes:: Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.
External Links:: MathReview, ZentralBlatt
Referenced by:: §25.10(i), §27.11, §27.2(i).
Encodings:: BibTeX

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C. de la Vallée Poussin (1896b) Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire

Mx+N

. Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

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Notes:: Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.
External Links:: MathReview, ZentralBlatt
Referenced by:: §25.10(i), §27.11, §27.2(i).
Encodings:: BibTeX

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G. Doetsch (1955) Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).

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External Links:: MathReview (I. I. Hirschman), ZentralBlatt
Referenced by:: §2.5(i).
Encodings:: BibTeX

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B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:: For a numerical error in one of the zeros see Amos (1985).
External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:: (This reference has several typographical errors.)
External Links:: ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §13.21(ii), §13.21(iii), §13.21(iv).
Encodings:: BibTeX

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5: Bibliography B

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G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:: Document
Referenced by:: §33.22(iv).
Encodings:: BibTeX

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K. L. Bell and N. S. Scott (1980) 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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P. Boalch (2005) From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. London Math. Soc. (3) 90 (1), pp. 167–208.

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External Links:: ISSN 0024-6115, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.9(iii).
Encodings:: BibTeX

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R. W. Butler and A. T. A. Wood (2002) Laplace approximations for hypergeometric functions with matrix argument. Ann. Statist. 30 (4), pp. 1155–1177.

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External Links:: ISSN 0090-5364, Document, MathReview, ZentralBlatt
Referenced by:: §35.6(iv), §35.7(iv), §35.7(iv), §35.9.
Encodings:: BibTeX

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R. W. Butler and A. T. A. Wood (2003) Laplace approximation for Bessel functions of matrix argument. J. Comput. Appl. Math. 155 (2), pp. 359–382.

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External Links:: ISSN 0377-0427, Document, MathReview (S. Bhargava), ZentralBlatt
Referenced by:: §35.5(ii), §35.5(iii).
Encodings:: BibTeX

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6: Bibliography K

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R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:: ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt
Referenced by:: §18.26(v), §18.26(v).
Encodings:: BibTeX

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V. I. Krylov and N. S. Skoblya (1985) A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.

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Notes:: Translated from the Russian by George Yankovsky. Reprint of the 1977 translation
External Links:: MathReview, ZentralBlatt
Referenced by:: §3.5(viii).
Encodings:: BibTeX

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7: 11.6 Asymptotic Expansions

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11.6.3

\int_{0}^{z}\mathbf{K}_{0}\left(t\right)\,\mathrm{d}t-\frac{2}{\pi}(\ln\left(2% z\right)+\gamma)\sim\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k+1}\frac{(2k)!(2k-1% )!}{(k!)^{2}(2z)^{2k}},

|\operatorname{ph}z|\leq\pi-\delta

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Symbols:: $\gamma$ : Euler’s constant, $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $z$ : complex variable, $k$ : nonnegative integer and $\delta$ : arbitrary small positive constant
A&S Ref:: 12.1.32
Referenced by:: §11.6(i)
Permalink:: http://dlmf.nist.gov/11.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(i), §11.6 and Ch.11

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11.6.4

\int_{0}^{z}\mathbf{M}_{0}\left(t\right)\,\mathrm{d}t+\frac{2}{\pi}(\ln\left(2% z\right)+\gamma)\sim\frac{2}{\pi}\sum_{k=1}^{\infty}\frac{(2k)!(2k-1)!}{(k!)^{% 2}(2z)^{2k}},

|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta

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Symbols:: $\gamma$ : Euler’s constant, $\mathbf{M}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $z$ : complex variable, $k$ : nonnegative integer and $\delta$ : arbitrary small positive constant
A&S Ref:: 12.2.8
Referenced by:: §11.6(i)
Permalink:: http://dlmf.nist.gov/11.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(i), §11.6 and Ch.11

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c_{3}(\lambda)=20\lambda^{-6}-4\lambda^{-4},

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8: Bibliography O

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F. Oberhettinger and L. Badii (1973) Tables of Laplace Transforms. Springer-Verlag, Berlin-New York.

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Notes:: Table errata: Math. Comp. v. 50 (1988), no. 182, pp. 653–654.
External Links:: ISBN 0-387-06350-1, MathReview, ZentralBlatt
Referenced by:: §1.14(viii), §10.22(vi), §10.43(vi), §11.5(iii), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(ii), §13.23(i), §14.17(v), §15.14, §18.17(ix), §20.10(ii), §5.13, §6.14(iii), §6.7(iv), §7.14(iii), §7.7(iii), §8.6(iii).
Encodings:: BibTeX

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S. Okui (1974) Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.

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External Links:: ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §10.22(vi), §10.43(vi).
Encodings:: BibTeX

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S. Okui (1975) Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.

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External Links:: ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §10.22(vi), §10.43(vi).
Encodings:: BibTeX

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J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:: Document, MathReview (David L. Elliott), ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

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I. Olkin (1959) A class of integral identities with matrix argument. Duke Math. J. 26 (2), pp. 207–213.

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External Links:: Document, MathReview (R. Bellman), ZentralBlatt
Referenced by:: §35.3(ii).
Encodings:: BibTeX

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9: Bibliography G

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W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:: For remark see same journal 24(1998), p. 355.
External Links:: Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt
Referenced by:: 2nd item, §3.5(v), §3.5(v).
Encodings:: BibTeX

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M. Geller and E. W. Ng (1969) A table of integrals of the exponential integral. J. Res. Nat. Bur. Standards Sect. B 73B, pp. 191–210.

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External Links:: MathReview (John Todd), ZentralBlatt
Referenced by:: §6.14(iii), §6.17, §6.7(iv).
Encodings:: BibTeX

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A. G. Gibbs (1973) Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.

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Notes:: Problem proposed by P. Smith.
External Links:: Document
Referenced by:: (9.10.14), (9.10.15), (9.10.16), §9.10(v).
Encodings:: BibTeX

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A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

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Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:: ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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10: 14.30 Spherical and Spheroidal Harmonics

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14.30.8

\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\overline{Y_{{l_{1}},{m_{1}}}\left(\theta,% \phi\right)}Y_{{l_{2}},{m_{2}}}\left(\theta,\phi\right)\sin\theta\,\mathrm{d}% \theta\,\mathrm{d}\phi=\delta_{l_{1},l_{2}}\delta_{m_{1},m_{2}}.

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $m$ : integer and $l$ : nonnegative integer
Referenced by:: §14.30(ii)
Permalink:: http://dlmf.nist.gov/14.30.E8
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.19):: As a notational clarification, the first spherical harmonic in the integral over $\theta$ been explicity marked up as a complex conjugate.
See also:: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

►See also (34.3.22), and for further related integrals see Askey et al. (1986). … ►As an example, Laplace’s equation

\nabla^{2}W=0

in spherical coordinates (§1.5(ii)): …