applied to generalized hypergeometric functions

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21: 10.59 Integrals

§10.59 Integrals

►

10.59.1

\int_{-\infty}^{\infty}e^{ibt}\mathsf{j}_{n}\left(t\right)\,\mathrm{d}t=\begin% {cases}\pi i^{n}P_{n}\left(b\right),&-1<b<1,\\ \frac{1}{2}\pi(\pm\mathrm{i})^{n},&b=\pm 1,\\ 0,&\pm b>1,\end{cases}

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Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind and $n$ : integer
A&S Ref:: 11.4.26
Referenced by:: §10.59
Permalink:: http://dlmf.nist.gov/10.59.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.59 and Ch.10

… ►For an integral representation of the Dirac delta in terms of a product of spherical Bessel functions of the first kind see §1.17(ii), and for a generalization see Maximon (1991). …

22: Bibliography

… ►

G. Alefeld and J. Herzberger (1983) Introduction to Interval Computations. Computer Science and Applied Mathematics, Academic Press Inc., New York.

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Notes:: Translated from the German by Jon Rokne
External Links:: ISBN 0-12-049820-0, MathReview (H. Ratschek), ZentralBlatt
Referenced by:: §3.1(ii).
Encodings:: BibTeX

… ►

G. Allasia and R. Besenghi (1989) Numerical Calculation of the Riemann Zeta Function and Generalizations by Means of the Trapezoidal Rule. In Numerical and Applied Mathematics, Part II (Paris, 1988), C. Brezinski (Ed.), IMACS Ann. Comput. Appl. Math., Vol. 1, pp. 467–472.

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

… ►

F. M. Arscott (1964b) Periodic Differential Equations. An Introduction to Mathieu, Lamé, and Allied Functions. International Series of Monographs in Pure and Applied Mathematics, Vol. 66, Pergamon Press, The Macmillan Co., New York.

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §28.1, §28.1, §28.10(i), §28.11, §28.12(ii), §28.17, §28.2(ii), §28.2(iii), §28.2(iv), §28.28(i), §28.28(ii), §28.29(i), §28.32(i), §28.32(ii), §28.4(i), §28.5(i), Ch.28, §29.1, §29.11, §29.12(i), §29.12(ii), §29.12(iii), §29.14, §29.15(ii), §29.15(ii), §29.17(i), §29.18(iii), Ch.29, §30.1, §30.10, §30.11(i), §30.11(vi), §30.2(ii), §30.4(iii), §30.6, §30.8(i), §30.8(ii), Ch.30, §31.4, §31.5, §31.9(ii), Notations F, Notations G, Notations M, Notations N.
Encodings:: BibTeX

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R. Askey and J. Wilson (1985) Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.

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External Links:: ISSN 0065-9266, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §18.21(ii), §18.22(ii), (18.28.24), §18.28(ii), Ch.18, Profile Richard A. Askey.
Encodings:: BibTeX

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R. Askey (1975b) Orthogonal Polynomials and Special Functions. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 21, Society for Industrial and Applied Mathematics, Philadelphia, PA.

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External Links:: ISBN 0-89871-018-9, MathReview (G. Gasper), ZentralBlatt
Referenced by:: §18.10(ii), §18.18(vii), Profile Richard A. Askey.
Encodings:: BibTeX

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23: 13.2 Definitions and Basic Properties

… ►

M\left(a,b,z\right)

is entire in

z

and

a

, and is a meromorphic function of

b

. … ►Although

M\left(a,b,z\right)

does not exist when

b=-n

n=0,1,2,\dots

, many formulas containing

M\left(a,b,z\right)

continue to apply in their limiting form. … ►In general,

U\left(a,b,z\right)

has a branch point at

z=0

. … ►Unless specified otherwise, however,

U\left(a,b,z\right)

is assumed to have its principal value. … ►

Kummer’s Transformations

…

24: Bibliography W

… ►

G. N. Watson (1910) The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

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External Links:: ZentralBlatt
Referenced by:: §15.8(v), §15.8(v).
Encodings:: BibTeX

… ►

G. Wei and B. E. Eichinger (1993) Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules. Ann. Inst. Statist. Math. 45 (3), pp. 467–475.

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External Links:: ISSN 0020-3157, Document, MathReview (N. K. Thakare), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

… ►

F. J. W. Whipple (1927) Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.

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External Links:: Document, ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

… ►

E. P. Wigner (1959) Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Pure and Applied Physics. Vol. 5, Academic Press, New York.

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Notes:: Expanded and improved ed. Translated from the German by J. J. Griffin.
External Links:: ISBN 0-12-750550-4, MathReview (A. S. Wightman), ZentralBlatt
Referenced by:: §34.12.
Encodings:: BibTeX

… ►

E. M. Wright (1940a) The asymptotic expansion of the generalized hypergeometric function. Proc. London Math. Soc. (2) 46, pp. 389–408.

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External Links:: ISSN 0024-6115, Document, MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §16.11(ii).
Encodings:: BibTeX

…

25: 18.5 Explicit Representations

… ►

Chebyshev

… ►

§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

… ►

Laguerre

… ►

Hermite

… ►For corresponding formulas for Chebyshev, Legendre, and the Hermite

\mathit{He}_{n}

polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). …

26: Bibliography N

… ►

M. Nardin, W. F. Perger, and A. Bhalla (1992a) 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

… ►

National Bureau of Standards (1958) Integrals of Airy Functions. National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:: MathReview (J. C. P. Miller), ZentralBlatt
Referenced by:: 2nd item, 3rd item.
Encodings:: BibTeX

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National Bureau of Standards (1967) Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:: MathReview, ZentralBlatt
Referenced by:: §28.1, §28.26(i), 6th item, Notations S, Notations S.
Encodings:: BibTeX

… ►

G. Nemes (2013c) Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.

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External Links:: ISSN 1065-2469, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt
Referenced by:: §5.11(i), §5.11(i).
Encodings:: BibTeX

… ►

N. E. Nørlund (1955) Hypergeometric functions. Acta Math. 94, pp. 289–349.

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External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §16.10, §16.8(ii), §16.8(ii).
Encodings:: BibTeX

…

27: Bibliography G

… ►

B. Gabutti (1980) On the generalization of a method for computing Bessel function integrals. J. Comput. Appl. Math. 6 (2), pp. 167–168.

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External Links:: ISSN 0771-050X, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

W. Gautschi (1998) The incomplete gamma functions since Tricomi. In Tricomi’s Ideas and Contemporary Applied Mathematics (Rome/Turin, 1997), Atti Convegni Lincei, Vol. 147, pp. 203–237.

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External Links:: FullText, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: Ch.8.
Encodings:: BibTeX

… ►

W. Gautschi (2002b) Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.

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External Links:: ISSN 0377-0427, Document, MathReview (Hasan Taşeli), ZentralBlatt
Referenced by:: §13.29(iii), §15.19(iii).
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2007a) Numerical Methods for Special Functions. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 978-0-898716-34-4, MathReview Entry, ZentralBlatt
Referenced by:: §3.10(iii), §3.11(ii), §3.5(v), §3.6(iii), §3.8(iii), §3.8(v), Ch.3, Profile Amparo Gil, Profile Javier Segura, Profile Nico M. Temme.
Encodings:: BibTeX

… ►

X. Guan, O. Zatsarinny, K. Bartschat, B. I. Schneider, J. Feist, and C. J. Noble (2007) General approach to few-cycle intense laser interactions with complex atoms. Phys. Rev. A 76, pp. 053411.

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External Links:: Document, Link
Referenced by:: §3.2(vi).
Encodings:: BibTeX

…

28: Bibliography R

… ►

W. H. Reid (1974b) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.

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External Links:: MathReview (T. W. Kao), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

… ►

F. E. Relton (1965) Applied Bessel Functions. Dover Publications Inc., New York.

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Notes:: Reprint of original Blackie & Son Limited, London, 1946.
External Links:: MathReview, ZentralBlatt
Referenced by:: §10.73(iii).
Encodings:: BibTeX

… ►

D. St. P. Richards (Ed.) (1992) Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications. Contemporary Mathematics, Vol. 138, American Mathematical Society, Providence, RI.

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External Links:: ISBN 0-8218-5159-4, MathReview, ZentralBlatt
Referenced by:: Ch.35, Profile Donald St. P. Richards.
Encodings:: BibTeX

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D. St. P. Richards (2004) Total positivity properties of generalized hypergeometric functions of matrix argument. J. Statist. Phys. 116 (1-4), pp. 907–922.

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External Links:: ISSN 0022-4715, Document, MathReview (Stamatis Koumandos), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

… ►

Hans-J. Runckel (1971) On the zeros of the hypergeometric function. Math. Ann. 191 (1), pp. 53–58.

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External Links:: ISSN 0025-5831, Document, MathReview (R. C. Varma), ZentralBlatt
Referenced by:: §15.13.
Encodings:: BibTeX

…

29: Bibliography B

… ►

W. N. Bailey (1928) Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.

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External Links:: Document, ZentralBlatt
Referenced by:: §16.12.
Encodings:: BibTeX

►

W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.

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External Links:: Document, ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

… ►

W. N. Bailey (1964) Generalized Hypergeometric Series. Stechert-Hafner, Inc., New York.

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External Links:: MathReview
Referenced by:: §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §17.1, §17.4(i), Ch.17.
Encodings:: BibTeX

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W. Bühring (1988) An analytic continuation formula for the generalized hypergeometric function. SIAM J. Math. Anal. 19 (5), pp. 1249–1251.

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External Links:: ISSN 0036-1410, Document, MathReview (H. M. Srivastava), ZentralBlatt
Referenced by:: §16.8(ii).
Encodings:: BibTeX

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W. Bühring (1992) Generalized hypergeometric functions at unit argument. Proc. Amer. Math. Soc. 114 (1), pp. 145–153.

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External Links:: ISSN 0002-9939, FullText, MathReview (R. K. Raina), ZentralBlatt
Referenced by:: §16.4(ii), §16.8(ii).
Encodings:: BibTeX

…

30: Bibliography C

… ►

L. G. Cabral-Rosetti and M. A. Sanchis-Lozano (2000) Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams. J. Comput. Appl. Math. 115 (1-2), pp. 93–99.

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External Links:: ISSN 0377-0427, Document, MathReview (Lorenzo L. Salcedo), ZentralBlatt
Referenced by:: §16.24(ii).
Encodings:: BibTeX

… ►

B. C. Carlson (1977b) Special Functions of Applied Mathematics. Academic Press, New York.

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External Links:: ISBN 0-12-160150-1, MathReview, ZentralBlatt
Referenced by:: (19.16.2_5), §19.16(ii), §19.16(ii), §19.16(iii), §19.18(i), §19.18(ii), §19.2(iv), §19.20(i), §19.20(iv), §19.20(v), §19.21(ii), §19.21(ii), §19.23, §19.25(i), §19.26(i), §19.26(iii), §19.28, §19.30(i), §19.30(ii), §19.30(iii), §19.32, §19.33(i), §19.33(ii), §19.33(iv), §19.34, Ch.19, §5.12, §5.6(ii), Ch.5, Profile Bille C. Carlson, Erratum (V1.0.18) for Equation (19.16.9).
Encodings:: BibTeX

… ►

F. Chapeau-Blondeau and A. Monir (2002) Numerical evaluation of the Lambert

W

function and application to generation of generalized Gaussian noise with exponent 1/2. IEEE Trans. Signal Process. 50 (9), pp. 2160–2165.

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External Links:: ISSN 1053-587X, Document, MathReview (J. Borwein), ZentralBlatt
Referenced by:: §4.44, §4.45(iii).
Encodings:: BibTeX

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M. A. Chaudhry and S. M. Zubair (1994) Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 55 (1), pp. 99–124.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §8.16.
Encodings:: BibTeX

… ►

S. Conde and S. L. Kalla (1981) On zeros of the hypergeometric function. Serdica 7 (3), pp. 243–249.

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External Links:: ISSN 0204-4140, MathReview (R. G. Buschman), ZentralBlatt
Referenced by:: §15.13.
Encodings:: BibTeX

…