integral%20representations

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11: Bibliography R

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M. Rahman (1981) A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:: ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.18(v).
Encodings:: BibTeX

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J. Raynal (1979) On the definition and properties of generalized

6

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

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W. H. Reid (1995) 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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W. H. Reid (1997a) 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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W. H. Reid (1997b) 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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12: 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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Á. Baricz and T. K. Pogány (2013) Integral representations and summations of the modified Struve function. Acta Math. Hungar. 141 (3), pp. 254–281.

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External Links:: ISSN 0236-5294, Document, FullText, MathReview (Eugenia N. Petropoulou), ZentralBlatt
Referenced by:: §11.7(v), §11.7(v).
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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W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:: Document, ZentralBlatt
Referenced by:: §10.73(iv).
Encodings:: BibTeX

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P. L. Butzer and M. Hauss (1992) Riemann zeta function: Rapidly converging series and integral representations. Appl. Math. Lett. 5 (2), pp. 83–88.

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External Links:: ISSN 0893-9659, Document, MathReview (Jacek Fabrykowski), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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13: Bibliography G

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A. Gil, J. Segura, and N. M. Temme (2004c) Integral representations for computing real parabolic cylinder functions. Numer. Math. 98 (1), pp. 105–134.

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External Links:: ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt
Referenced by:: §12.18.
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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H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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External Links:: FullText, MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

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K. I. Gross and R. A. Kunze (1976) Bessel functions and representation theory. I. J. Functional Analysis 22 (2), pp. 73–105.

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External Links:: Document, MathReview (E. A. Thieleker), ZentralBlatt
Referenced by:: §35.5(i).
Encodings:: BibTeX

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E. Grosswald (1985) Representations of Integers as Sums of Squares. Springer-Verlag, New York.

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External Links:: ISBN 0-387-96126-7, MathReview (Harvey Cohn), ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

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14: Bibliography P

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R. B. Paris and S. Cang (1997) An asymptotic representation for

\zeta(\frac{1}{2}+it)

. Methods Appl. Anal. 4 (4), pp. 449–470.

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External Links:: ISSN 1073-2772, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.9, §8.22(ii).
Encodings:: BibTeX

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J. Patera and P. Winternitz (1973) A new basis for the representation of the rotation group. Lamé and Heun polynomials. J. Mathematical Phys. 14 (8), pp. 1130–1139.

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External Links:: Document, ZentralBlatt
Referenced by:: §29.18(iv).
Encodings:: BibTeX

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B. Pichon (1989) Numerical calculation of the generalized Fermi-Dirac integrals. Comput. Phys. Comm. 55 (2), pp. 127–136.

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External Links:: Document
Referenced by:: §25.18(i).
Encodings:: BibTeX

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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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W. H. Press and S. A. Teukolsky (1990) Elliptic integrals. Computers in Physics 4 (1), pp. 92–96.

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Referenced by:: 2nd item.
Encodings:: BibTeX

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15: 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},

… ►These and higher coefficients

c_{k}(\lambda)

can be computed via the representations in Nemes (2015b). …

16: 14.30 Spherical and Spheroidal Harmonics

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Explicit Representation

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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). … ►For a series representation of the product of two Dirac deltas in terms of products of spherical harmonics see §1.17(iii). …