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11: Bibliography G

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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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Notes:

Information on Mersenne primes and the source code for the Great Internet Mersenne Prime Search.

External Links:

Home Page

Referenced by:

5th item.

Encodings:

BibTeX

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Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.

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External Links:

ISSN 0026-9255, MathReview (Peter Stevenhagen), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Number-Divisor Tables. British Association Mathematical Tables, Vol. VIII, Cambridge University Press, Cambridge, England.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§27.21.

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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12: Bibliography L

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Ratios of Bessel functions and roots of $\alpha J_{\nu }(x)+x{J}_{\nu }^{\prime }(x)=0$ . J. Math. Anal. Appl. 240 (1), pp. 174–204.

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External Links:

ISSN 0022-247X, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.14, §10.15, §10.21(xii).

Encodings:

BibTeX

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Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§4.13, 2nd item, §4.48(iv).

Encodings:

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An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

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Guide to Tables in the Theory of Numbers. Bulletin of the National Research Council, No. 105, National Research Council, Washington, D.C..

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Notes:

Reprinted in 1961.

External Links:

MathReview (R. D. Carmichael), ZentralBlatt

Referenced by:

§27.20, §27.20, §27.21, §27.21.

Encodings:

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List of Prime Numbers from 1 to 10,006,721. Publ. No. 165, Carnegie Institution of Washington, Washington, D.C..

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External Links:

ZentralBlatt

Referenced by:

§27.21.

Encodings:

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13: Software Index

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	Open Source													With Book						Commercial
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9.20(ii) $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$, $x\in ℝ$	✓	✓	✓	✓	✓	✓		✓	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓
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20 Theta Functions
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24.21(ii) $B_n$, $B_n\left(x\right)$, $E_n$, $E_n\left(x\right)$	✓	✓						✓		✓		a	✓	✓			✓		✓		✓	✓	✓		Derive, MuPAD
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27 Functions of Number Theory
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An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

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14: 9.7 Asymptotic Expansions

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9.7.3 $$\chi (x)\equiv {\pi }^{1/2}\mathrm{\Gamma }\left(\frac{1}{2}x+1\right)/\mathrm{\Gamma }\left(\frac{1}{2}x+\frac{1}{2}\right).$$

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Defines:

$\chi (x)$: function (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter and $\equiv$: equals by definition

Source:

Olver (1997b, (13.02), p. 225)

Referenced by:

Erratum (V1.0.17) for Equations (9.7.3), (9.7.4)

Permalink:

http://dlmf.nist.gov/9.7.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.0.17):

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 $x$.

See also:

Annotations for §9.7(i), §9.7 and Ch.9

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9.7.4 $$\chi (x)\sim {(\frac{1}{2}\pi x)}^{1/2}.$$

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter and $\chi (x)$: function

Source:

Olver (1997b, p. 225)

Referenced by:

Erratum (V1.0.17) for Equations (9.7.3), (9.7.4)

Permalink:

http://dlmf.nist.gov/9.7.E4

Encodings:

pMML, png, TeX

Modification (effective with 1.0.17):

Originally the formula 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 $x$.

See also:

Annotations for §9.7(i), §9.7 and Ch.9

►Numerical values of $\chi (n)$ are given in Table 9.7.1 for $n=1(1)20$ to 2D. … ►

$|{\mathrm{Ai}}^{\prime }\left(x\right)|\le {\displaystyle \frac{x^{1/4}{\mathrm{e}}^{-\xi }}{2\sqrt{\pi }}}\left(1+{\displaystyle \frac{7}{72\xi }}\right)$ ,

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$${\mathrm{Bi}}^{\prime }\left(x\right)\le \frac{x^{1/4}{\mathrm{e}}^{\xi }}{\sqrt{\pi }}\left(1+\left(\frac{\pi }{2}+1\right)\frac{7}{72\xi }\right),$$

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15: 25.6 Integer Arguments

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§25.6(i) Function Values

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25.6.2 $$\zeta \left(2n\right)=\frac{{(2\pi )}^{2n}}{2(2n)!}\left|B_{2n}\right|,$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: nonnegative integer

Sources:

Magnus et al. (1966, p. 19); Apostol (1976, (22), p. 266)

A&S Ref:

23.2.16

Referenced by:

(25.11.21)

Permalink:

http://dlmf.nist.gov/25.6.E2

Encodings:

pMML, png, TeX

See also:

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

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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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25.6.11 $${\zeta }^{\prime }\left(0\right)=-\frac{1}{2}\ln \left(2\pi \right).$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $\ln z$: principal branch of logarithm function

Source:

Apostol (1985a, p. 223)

A&S Ref:

23.2.13

Permalink:

http://dlmf.nist.gov/25.6.E11

Encodings:

pMML, png, TeX

See also:

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

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25.6.15 $${\zeta }^{\prime }\left(2n\right)=\frac{{(-1)}^{n+1}{(2\pi )}^{2n}}{2(2n)!}\left(2n{\zeta }^{\prime }\left(1-2n\right)-(\psi \left(2n\right)-\ln \left(2\pi \right))B_{2n}\right).$$

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Source:

Miller and Adamchik (1998, (2), p. 202)

Referenced by:

(25.11.21)

Permalink:

http://dlmf.nist.gov/25.6.E15

Encodings:

pMML, png, TeX

See also:

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

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16: 9.9 Zeros

… ►On the real line, $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ each have an infinite number of zeros, all of which are negative. … ► $\mathrm{Ai}\left(z\right)$ and ${\mathrm{Ai}}^{\prime }\left(z\right)$ have no other zeros. However, $\mathrm{Bi}\left(z\right)$ and ${\mathrm{Bi}}^{\prime }\left(z\right)$ each have an infinite number of complex zeros. … ►Tables 9.9.1 and 9.9.2 give 10D values of the first ten real zeros of $\mathrm{Ai}$, ${\mathrm{Ai}}^{\prime }$, $\mathrm{Bi}$, ${\mathrm{Bi}}^{\prime }$, together with the associated values of the derivative or the function. Tables 9.9.3 and 9.9.4 give the corresponding results for the first ten complex zeros of $\mathrm{Bi}$ and ${\mathrm{Bi}}^{\prime }$ in the upper half plane. …

17: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

Encodings:

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Hill’s Equation. Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney.

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Notes:

Reprinted by Dover Publications, Inc., New York, 1979.

External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§1.3(iii), §28.29(i), §28.29(ii), §28.29(ii), §28.29(iii), §28.29(iii), §28.29(iii), §28.30(i), Ch.28, §29.3(i), §29.9.

Encodings:

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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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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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External Links:

FullText

Referenced by:

§8.24(i).

Encodings:

BibTeX

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

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18: Bibliography W

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The irregular primes to $125000$ . Math. Comp. 32 (142), pp. 583–591.

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External Links:

FullText, Document, MathReview (Wells Johnson), ZentralBlatt

Referenced by:

§24.20.

Encodings:

BibTeX

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Prime Divisors of the Bernoulli and Euler Numbers. In Number Theory for the Millennium, III (Urbana, IL, 2000), pp. 357–374.

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External Links:

FullText, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.20.

Encodings:

BibTeX

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:

ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

Encodings:

BibTeX

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Generating functions of class-numbers. Compositio Math. 1, pp. 39–68.

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External Links:

ISSN 0010-437X, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§20.7(v), §20.8.

Encodings:

BibTeX

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Some explicit Padé approximants for the function ${\mathrm{\Phi }}^{\prime }/\mathrm{\Phi }$ and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal. 16 (4), pp. 887–895.

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External Links:

ISSN 0036-1410, Document, MathReview (Wyman Fair), ZentralBlatt

Referenced by:

§13.31(ii).

Encodings:

BibTeX

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19: Bibliography D

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Multiplicative Number Theory. 3rd edition, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York.

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Notes:

Revised and with a preface by Hugh L. Montgomery

External Links:

ISBN 0-387-95097-4, MathReview, ZentralBlatt

Referenced by:

§27.12.

Encodings:

BibTeX

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Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.

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External Links:

ISSN 0022-314X, Document, MathReview (Wen Peng Zhang), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

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Bernoulli Numbers and Confluent Hypergeometric Functions. In Number Theory for the Millennium, I (Urbana, IL, 2000), pp. 343–363.

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External Links:

ISBN 1-56881-126-8, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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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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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

External Links:

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

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

BibTeX

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20: Bibliography P

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Notes:

Computer algebra system designed for computations in number theory. Many transcendental functions are also included. PARI is also available as a C library to allow for faster computations. Originally developed by Henri Cohen and co-workers at the Université Bordeaux I.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

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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Voronoi type congruences for Bernoulli numbers. In Voronoi’s Impact on Modern Science. Book I, P. Engel and H. Syta (Eds.),

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External Links:

ZentralBlatt

Referenced by:

§24.10(iii).

Encodings:

BibTeX

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Notes:

Information on primes, primality testing, and factorization including links to programs and lists of primes.

External Links:

Home Page

Referenced by:

7th item.

Encodings:

BibTeX

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