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11: 33.23 Methods of Computation

… ►Noble (2004) obtains double-precision accuracy for $W_{-\eta ,\mu }\left(2\rho \right)$ for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …

12: 27.5 Inversion Formulas

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27.5.6 $$G(x)=\sum _{n\le x}F\left(\frac{x}{n}\right)⟺F(x)=\sum _{n\le x}\mu \left(n\right)G\left(\frac{x}{n}\right),$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer, $x$: real number, $F(s)$: Dirichlet series and $G(s)$: Dirichlet series

A&S Ref:

24.3.1 II.C

Permalink:

http://dlmf.nist.gov/27.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

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27.5.7 $$G(x)=\sum _{m=1}^{\mathrm{\infty }}\frac{F(mx)}{m^s}⟺F(x)=\sum _{m=1}^{\mathrm{\infty }}\mu \left(m\right)\frac{G(mx)}{m^s},$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $m$: positive integer, $x$: real number, $F(s)$: Dirichlet series and $G(s)$: Dirichlet series

Referenced by:

§27.17, §27.5

Permalink:

http://dlmf.nist.gov/27.5.E7

Encodings:

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See also:

Annotations for §27.5 and Ch.27

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13: Bibliography P

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A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.

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Notes:

Extended-precision Fortran, maximum accuracy 12D.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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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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Chebyshev series approximations for the zeros of the Bessel functions. J. Comput. Phys. 53 (1), pp. 188–192.

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External Links:

ISSN 0021-9991, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.76(ii).

Encodings:

BibTeX

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Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt

Referenced by:

§1.14(vii).

Encodings:

BibTeX

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Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 1 (4), pp. 372–379.

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External Links:

ISSN 0098-3500, Document, GAMS (module 498; package TOMS), ZentralBlatt

Referenced by:

1st item, 4th item.

Encodings:

BibTeX

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14: Bibliography D

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Chebyshev expansion of the associated Legendre polynomial $P_L^M(x)$ . Comput. Phys. Comm. 18 (1), pp. 63–71.

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Notes:

Double-precision Fortran, maximum accuracy 25D

External Links:

Document, Code

Referenced by:

3rd item.

Encodings:

BibTeX

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Chebyshev series for the spherical Bessel function $j_l(r)$ . Comput. Phys. Comm. 18 (1), pp. 73–86.

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External Links:

Document

Referenced by:

§10.76(iii).

Encodings:

BibTeX

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The Taylor Series. Oxford University Press, Oxford.

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Notes:

Reprinted by Dover Publications Inc., New York, 1957

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.9(iii).

Encodings:

BibTeX

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Bessel functions $J_n(z)$ and $Y_n(z)$ of integer order and complex argument. Comput. Phys. Comm. 78 (1-2), pp. 181–189.

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Notes:

Double-precision Pascal, maximum accuracy 7D.

External Links:

ISSN 0010-4655, Document, Code, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

2nd item.

Encodings:

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Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point. SIAM J. Math. Anal. 21 (6), pp. 1594–1618.

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External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.26, §2.8(vi).

Encodings:

BibTeX

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15: Bibliography H

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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

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Divergent Series. Clarendon Press, Oxford.

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Notes:

Reprinted by the American Mathematical Society in 2000.

External Links:

MathReview (R. P. Agnew), ZentralBlatt

Referenced by:

§1.15(ii), §1.15(viii).

Encodings:

BibTeX

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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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Note on some hypergeometric series of higher order. J. London Math. Soc. 4, pp. 50–54.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.9.

Encodings:

BibTeX

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Two-center Coulomb functions. Comput. Phys. Comm. 103 (2-3), pp. 209–216.

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Notes:

Double-precision Fortran code.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

Encodings:

BibTeX

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16: 16.15 Integral Representations and Integrals

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16.15.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)\mathrm{\Gamma }\left(\gamma -\beta -{\beta }^{\prime }\right)}{\iint }_{\mathrm{\Delta }}\frac{u^{\beta -1}{v}^{{\beta }^{\prime }-1}{(1-u-v)}^{\gamma -\beta -{\beta }^{\prime }-1}}{{(1-ux)}^{\alpha }{(1-vy)}^{{\alpha }^{\prime }}}dudv,$$ $\mathrm{\Re }\left(\gamma -\beta -{\beta }^{\prime }\right)>0$, $\mathrm{\Re }\beta >0$, $\mathrm{\Re }{\beta }^{\prime }>0$,

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Symbols:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Notes:

Erdélyi et al. (1953a, 5.8.1(3)) contains a typo.

Referenced by:

Erratum (V1.0.14) for Equation (16.15.3), Erratum (V1.0.14) for Equation (16.15.3)

Permalink:

http://dlmf.nist.gov/16.15.E3

Encodings:

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Clarification (effective with 1.0.14):

A stray “$f$” was removed; it appeared after the comma at the end of this equation.

See also:

Annotations for §16.15 and Ch.16

… ►For these and other formulas, including double Mellin–Barnes integrals, see Erdélyi et al. (1953a, §5.8). These representations can be used to derive analytic continuations of the Appell functions, including convergent series expansions for large $x$, large $y$, or both. …

17: 20.5 Infinite Products and Related Results

… ►With the given conditions the infinite series in (20.5.10)–(20.5.13) converge absolutely and uniformly in compact sets in the $z$-plane. … ►

§20.5(iii) Double Products

… ►These double products are not absolutely convergent; hence the order of the limits is important. …

18: Bibliography B

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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

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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:

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The double confluent Heun equation: Characteristic exponent and connection formulae. Methods Appl. Anal. 1 (3), pp. 348–370.

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External Links:

ISSN 1073-2772, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§31.12.

Encodings:

BibTeX

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Expansions of Appell’s double hypergeometric functions. Quart. J. Math., Oxford Ser. 11, pp. 249–270.

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External Links:

Document, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§15.16, (16.15.4), §16.15, §16.16(i).

Encodings:

BibTeX

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Expansions of Appell’s double hypergeometric functions. II. Quart. J. Math., Oxford Ser. 12, pp. 112–128.

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External Links:

Document, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§16.16(i).

Encodings:

BibTeX

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19: 35.10 Methods of Computation

… ►Other methods include numerical quadrature applied to double and multiple integral representations. … ►Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …

20: 3.11 Approximation Techniques

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§3.11(ii) Chebyshev-Series Expansions

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Summation of Chebyshev Series: Clenshaw’s Algorithm

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Complex Variables

… ►be a formal power series. …