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11: 8.26 Tables

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§8.26(ii) Incomplete Gamma Functions

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

12: Bibliography S

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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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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

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Notes:

Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.

External Links:

ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt

Referenced by:

§3.3(vi), §3.4(i), §3.5(i).

Encodings:

BibTeX

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13: Bibliography D

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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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14: Peter L. Walker

… ►, where he was a founding member. … ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. … ►

20 Theta Functions

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22 Jacobian Elliptic Functions

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23 Weierstrass Elliptic and Modular Functions

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15: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

… ►The function $\zeta (s,a)$ was introduced in Hurwitz (1882) and defined by the series expansion … ►As a function of $a$, with $s$ ($\ne 1$) fixed, $\zeta (s,a)$ is analytic in the half-plane $\mathrm{\Re }a>0$. The Riemann zeta function is a special case: … ►

§25.11(xii) $a$-Asymptotic Behavior

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16: Publications

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Toward a Revised NBS Handbook of Mathematical Functions, Technical Report NISTIR 6072 (September 1997), National Institute of Standards and Technology.

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The DLMF Project: A New Initiative in Classical Special Functions, in Special Functions—Proceedings of the International Workshop, Hong Kong, June 21-25, 1999 (C. Dunkl, M. Ismail, R. Wong, eds.), World Scientific, pp. 207–220.

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Handbook of Mathematical Functions, in A Century of Excellence in Measurements Standards and Technology (D. R. Lide, ed.), CRC Press, pp. 135–139.

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Technical Aspects of the Digital Library of Mathematical Functions, Annals of Mathematics and Artificial Intelligence—Special Issue on Mathematical Knowledge Management, Vol. 38, Nos. 1–3, pp. 121–136.

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NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53.

17: 8.17 Incomplete Beta Functions

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8.17.24 $$I_x(m,n)={(1-x)}^n\sum _{j=m}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{n+j-1}{j}\right)x^j,$$ $m,n$ positive integers; .

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Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

26.5.26 (The upper limit of summation has been corrected.)

Referenced by:

§8.17(i), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/8.17.E24

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation is the same as Equation (26.5.26) of Abramowitz and Stegun (1964), except that the upper limit of the summation has been corrected to $\mathrm{\infty }$.

Suggested 2011-03-23 by Stephen Bourn

See also:

Annotations for §8.17(vii), §8.17 and Ch.8

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18: 7.24 Approximations

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§7.24(i) Approximations in Terms of Elementary Functions

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Hastings (1955) gives several minimax polynomial and rational approximations for $\mathrm{erf}x$, $\mathrm{erfc}x$ and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$.

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Cody (1969) provides minimax rational approximations for $\mathrm{erf}x$ and $\mathrm{erfc}x$. The maximum relative precision is about 20S.

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Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

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Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for $F\left(z\right)$, $\mathrm{erf}z$, $\mathrm{erfc}z$, $C\left(z\right)$, and $S\left(z\right)$; approximate errors are given for a selection of $z$-values.

19: Bibliography K

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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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20: Bibliography B

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A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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A short table of the functions ${\mathrm{Ki}}_n(x)$, from $n=1$ to $n=16$ . Phil. Mag. Series 7 20, pp. 343–347.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.43(iii), 2nd item.

Encodings:

BibTeX

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