DLMF: Untitled Document

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M. Kodama (2008) Algorithm 877: A subroutine package for cylindrical functions of complex order and nonnegative argument. ACM Trans. Math. Software 34 (4), pp. Art. 22, 21.

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External Links:: ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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P. Koev and A. Edelman (2006) The efficient evaluation of the hypergeometric function of a matrix argument. Math. Comp. 75 (254), pp. 833–846.

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Notes:: For Matlab software for this function, see Plamen Koev’s home page.
External Links:: Home Page, ISSN 0025-5718, Document, MathReview (Luis Verde-Star), ZentralBlatt
Referenced by:: §35.10, 2nd item.
Encodings:: BibTeX

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K. S. Kölbig (1972c) Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument. Comput. Phys. Comm. 4, pp. 221–226.

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Notes:: Single-precision Fortran, accuracy 12-14S
External Links:: Document, Code
Referenced by:: 1st item.
Encodings:: BibTeX

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T. H. Koornwinder and I. Sprinkhuizen-Kuyper (1978) Hypergeometric functions of

2\times 2

matrix argument are expressible in terms of Appel’s functions

F_{4}

. Proc. Amer. Math. Soc. 70 (1), pp. 39–42.

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External Links:: FullText, MathReview (P. N. Rathie), ZentralBlatt
Referenced by:: §35.7(ii).
Encodings:: BibTeX

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P. Kravanja, O. Ragos, M. N. Vrahatis, and F. A. Zafiropoulos (1998) ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument. Comput. Phys. Comm. 113 (2-3), pp. 220–238.

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Notes:: Double-precision Fortran, maximum accuracy 25S.
External Links:: ISSN 0010-4655, Document, Code, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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D. E. Amos (1985) A subroutine package for Bessel functions of a complex argument and nonnegative order. Technical Report Technical Report SAND85-1018, Sandia National Laboratories, Albuquerque, NM.

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External Links:: FullText
Referenced by:: 1st item, B. Döring (1966).
Encodings:: BibTeX

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D. E. Amos (1986) Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order. ACM Trans. Math. Software 12 (3), pp. 265–273.

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Notes:: Single and double precision, maximum accuracy 18S.
External Links:: ISSN 0098-3500, Document, Remark #1. Remark #2. GAMS (module 644; package TOMS), MathReview Entry, ZentralBlatt
Referenced by:: 1st item, 1st item, 2nd item, 1st item.
Encodings:: BibTeX

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T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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External Links:: ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt
Referenced by:: (25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.
Encodings:: BibTeX

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R. W. B. Ardill and K. J. M. Moriarty (1978) Spherical Bessel functions

j_{n}

and

y_{n}

of integer order and real argument. Comput. Phys. Comm. 14 (3-4), pp. 261–265.

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Notes:: Single-precision Fortran, maximum accuracy 10S.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 1st item.
Encodings:: BibTeX

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F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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External Links:: Document, MathReview (I. Marx), ZentralBlatt
Referenced by:: §29.8.
Encodings:: BibTeX

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O. M. Ogreid and P. Osland (1998) Summing one- and two-dimensional series related to the Euler series. J. Comput. Appl. Math. 98 (2), pp. 245–271.

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External Links:: ISSN 0377-0427, Document, MathReview (Boris Petrovich Osilenker), ZentralBlatt
Referenced by:: §25.8.
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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A. B. Olde Daalhuis (1994) Asymptotic expansions for

q

-gamma,

q

-exponential, and

q

-Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.

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External Links:: ISSN 0022-247X, Document, MathReview (P. K. Banerji), ZentralBlatt
Referenced by:: §5.18(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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Equation (35.7.3)

Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${{}_{2}F_{1}}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.

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Equations (33.11.2)–(33.11.7)

The arguments of some of the functions in (33.11.2)–(33.11.7) were included to improve clarity of the presentation.

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Chapter 35 Functions of Matrix Argument

The generalized hypergeometric function of matrix argument ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ , was linked inadvertently as its single variable counterpart ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ . Furthermore, the Jacobi function of matrix argument $P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)$ , and the Laguerre function of matrix argument $L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)$ , were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by $P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)$ , and $L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)$ . In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

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Equations (9.7.3), (9.7.4)

Originally the function $\chi$ was presented with argument given by a positive integer $n$ . It has now been clarified to be valid for argument given by a positive real number $x$ .

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Chapter 25 Zeta and Related Functions

A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

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S. Makinouchi (1966) Zeros of Bessel functions

J_{\nu}\,(x)

and

Y_{\nu}\,(x)

accurate to twenty-nine significant digits. Technology Reports of the Osaka University 16 (685), pp. 1–44.

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Referenced by:: 6th item.
Encodings:: BibTeX

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J. McMahon (1894) On the roots of the Bessel and certain related functions. Ann. of Math. 9 (1-6), pp. 23–30.

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External Links:: ISSN 0003-486X, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §10.21(x).
Encodings:: BibTeX

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Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

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R. Morris (1979) The dilogarithm function of a real argument. Math. Comp. 33 (146), pp. 778–787.

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External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item, 4th item.
Encodings:: BibTeX

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M. E. Muldoon (1981) The variation with respect to order of zeros of Bessel functions. Rend. Sem. Mat. Univ. Politec. Torino 39 (2), pp. 15–25.

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External Links:: MathReview (S. Ahmed), ZentralBlatt
Referenced by:: §10.21(iv).
Encodings:: BibTeX

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F. Calogero (1978) Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial

{L}_{n}^{\alpha}(x)

as the index

\alpha\rightarrow\infty

and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

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External Links:: ISSN 0024-1318, Document, MathReview (R. A. Askey)
Referenced by:: §18.7(iii).
Encodings:: BibTeX

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J. B. Campbell (1979) Bessel functions

J_{\nu}(x)

and

Y_{\nu}(x)

of real order and real argument. Comput. Phys. Comm. 18 (1), pp. 133–142.

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Notes:: Double-precision Fortran, maximum accuracy 16S.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 4th item.
Encodings:: BibTeX

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J. B. Campbell (1981) Bessel functions

I_{\nu}(x)

and

K_{\nu}(x)

of real order and complex argument. Comput. Phys. Comm. 24 (1), pp. 97–105.

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Notes:: Double-precision Fortran, maximum accuracy 16S. For erratum see same journal 25 (1982), p. 207.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 2nd item.
Encodings:: BibTeX

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W. J. Cody (1991) Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

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External Links:: ISSN 0098-3500, Document, MathReview, ZentralBlatt
Referenced by:: §5.24(ii), §5.24(iii).
Encodings:: BibTeX

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J. P. Coleman (1980) A Fortran subroutine for the Bessel function

J_{n}(x)

of order

0

to

10

. Comput. Phys. Comm. 21 (1), pp. 109–118.

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Notes:: Double-precision Fortran, maximum accuracy 16S.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 4th item.
Encodings:: BibTeX

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relations to Bessel functions of matrix argument

11—16 of 16 matching pages

11: Bibliography K

12: Bibliography

13: Bibliography O

14: Errata

15: Bibliography M

16: Bibliography C