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21: William P. Reinhardt

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20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

22: 20.7 Identities

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§20.7(ii) Addition Formulas

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§20.7(iii) Duplication Formula

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§20.7(iv) Reduction Formulas for Products

… ►See Lawden (1989, pp. 19–20). … ►

§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

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23: 6.19 Tables

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Zhang and Jin (1996, pp. 652, 689) includes $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $x=0(.5)20(2)30$, 8D; $\mathrm{Ei}\left(x\right)$, $E_1\left(x\right)$, $x=[0,100]$, 8S.

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Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $z{\mathrm{e}}^z{E}_1\left(z\right)$, $x=-19(1)20$, $y=0(1)20$, 6D; ${\mathrm{e}}^z{E}_1\left(z\right)$, $x=-4(.5)-2$, $y=0(.2)1$, 6D; $E_1\left(z\right)+\ln z$, $x=-2(.5)2.5$, $y=0(.2)1$, 6D.

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Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_1\left(z\right)$, $\pm x=0.5,1,3,5,10,15,20,50,100$, $y=0(.5)1(1)5(5)30,50,100$, 8S.

24: Peter L. Walker

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20 Theta Functions

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25: Staff

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William P. Reinhardt, University of Washington, Chaps. 20, 22, 23

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Peter L. Walker, American University of Sharjah, Chaps. 20, 22, 23

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William P. Reinhardt, University of Washington, for Chaps. 20, 22, 23

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Peter L. Walker, American University of Sharjah, for Chaps. 20, 22, 23

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26: 25.6 Integer Arguments

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25.6.3 $$\zeta \left(-n\right)=-\frac{B_{n+1}}{n+1},$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function and $n$: nonnegative integer

Source:

Apostol (1976, (20), p. 266); with $n\ne 0$

A&S Ref:

23.2.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

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§25.6(ii) Derivative Values

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§25.6(iii) Recursion Formulas

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27: Bibliography G

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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

Encodings:

BibTeX

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Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, pp. 251–270.

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External Links:

ISSN 0025-5718, Document, Link, MathReview (R. E. Barnhill)

Referenced by:

§18.39(iii).

Encodings:

BibTeX

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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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Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

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External Links:

FullText, MathReview (B. M. Stewart), ZentralBlatt

Referenced by:

§24.15(iii), §24.6.

Encodings:

BibTeX

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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28: Bibliography I

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The zeros of regular Coulomb wave functions and of their derivatives. Math. Comp. 29, pp. 878–887.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§3.8(iii).

Encodings:

BibTeX

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

Encodings:

BibTeX

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The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).

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Notes:

In Russian. English translation: Math. USSR-Izv. 31(1988), no. 1, pp. 193–207

External Links:

ISSN 0373-2436, Document, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§32.11(iii).

Encodings:

BibTeX

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Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution. J. Phys. A 31 (17), pp. 4073–4113.

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External Links:

ISSN 0305-4470, Document, MathReview (Yoshitsugu Takei), ZentralBlatt

Referenced by:

§32.11(v).

Encodings:

BibTeX

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29: Bibliography N

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On simplified asymptotic formulas for a class of Mathieu functions. SIAM J. Math. Anal. 15 (6), pp. 1205–1213.

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External Links:

ISSN 0036-1410, Document, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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On a simplified asymptotic formula for the Mathieu function of the third kind. SIAM J. Math. Anal. 18 (6), pp. 1616–1629.

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External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

Encodings:

BibTeX

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

Encodings:

BibTeX

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30: 9.18 Tables

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Miller (1946) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$ for $x=-20(.01)2;$ ${\log }_{10}\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)/\mathrm{Ai}\left(x\right)$ for $x=0(.1)25(1)75$; $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=-10(.1)2.5$; ${\log }_{10}\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)/\mathrm{Bi}\left(x\right)$ for $x=0(.1)10$; $M\left(x\right)$, $N\left(x\right)$, $\theta \left(x\right)$, $\varphi \left(x\right)$ (respectively $F(x)$, $G(x)$, $\chi (x)$, $\psi (x)$) for $x=-80(1)-30(.1)0$. Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

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Zhang and Jin (1996, p. 337) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

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Miller (1946) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$. Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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Sherry (1959) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; 20S.

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Zhang and Jin (1996, p. 339) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$; 8D.

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