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11: 10.75 Tables

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Olver (1962) provides tables for the uniform asymptotic expansions given in §10.20(i), including $\zeta$ and ${(4\zeta /(1-x^2))}^{\frac{1}{4}}$ as functions of $x$ $(=z)$ and the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, $D_k(\zeta )$ as functions of $\zeta$. These enable $J_{\nu }\left(\nu x\right)$, $Y_{\nu }\left(\nu x\right)$, $J_{\nu }^{\prime }\left(\nu x\right)$, $Y_{\nu }^{\prime }\left(\nu x\right)$ to be computed to 10S when $\nu \ge 15$, except in the neighborhoods of zeros.

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Makinouchi (1966) tabulates all values of $j_{\nu ,m}$ and $y_{\nu ,m}$ in the interval $(0,100)$, with at least 29S. These are for $\nu =0(1)5$, 10, 20; $\nu =\frac{3}{2}$, $\frac{5}{2}$; $\nu =m/n$ with $m=1(1)n-1$ and $n=3(1)8$, except for $\nu =\frac{1}{2}$.

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Bickley et al. (1952) tabulates $x^{-n}{I}_n\left(x\right)$ or ${\mathrm{e}}^{-x}{I}_n\left(x\right)$, $x^n{K}_n\left(x\right)$ or ${\mathrm{e}}^x{K}_n\left(x\right)$, $n=2(1)20$, $x=0$(.01 or .1) 10(.1) 20, 8S; $I_n\left(x\right)$, $K_n\left(x\right)$, $n=0(1)20$, $x=0$ or $0.1(.1)20$, 10S.

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Leung and Ghaderpanah (1979), tabulates all zeros of the principal value of $K_n\left(z\right)$, for $n=2(1)10$, 29S.

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Olver (1960) tabulates $a_{n,m}^{\prime }$, ${𝗃}_n\left(a_{n,m}^{\prime }\right)$, $b_{n,m}^{\prime }$, ${𝗒}_n\left(b_{n,m}^{\prime }\right)$, $n=1(1)20$, $m=1(1)50$, 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to \mathrm{\infty }$.

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12: Errata

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17.6.1 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,c/(ab))=\frac{{(c/a,c/b;q)}_{\mathrm{\infty }}}{{(c,c/(ab);q)}_{\mathrm{\infty }}},$$

The constraint was added.

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18.28.2 $${\int }_{-1}^1{p}_n(x)p_m(x)w(x)dx=h_n{\delta }_{n,m},$$ $|a|,|b|,|c|,|d|\le 1$, $ab,ac,ad,bc,bd,cd\ne 1$

The constraint of this equation was updated to include $ab,ac,ad,bc,bd,cd\ne 1$.

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An entire new Subsection 13.8(iv) “Large $a$ and $b$”, was added.

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Version 1.0.9 (August 29, 2014)

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Version 1.0.3 (Aug 29, 2011)

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13: 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:

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Large orders and summability of eigenvalue perturbation theory: A mathematical overview. Int. J. Quantum Chem. 21, pp. 3–25.

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External Links:

Document

Referenced by:

§22.19(ii).

Encodings:

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Sturm oscillation and comparison theorems. In Sturm-Liouville theory, pp. 29–43.

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External Links:

Document, Link, MathReview Entry

Referenced by:

§18.36(vi), §18.39(i).

Encodings:

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A new Fortran program for the $9$-$j$ angular momentum coefficient. Comput. Phys. Comm. 56 (2), pp. 231–248.

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External Links:

Document, ZentralBlatt

Referenced by:

§34.13.

Encodings:

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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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14: Bibliography L

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A Numerical Library in C for Scientists and Engineers. CRC Press, Boca Raton, FL.

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Notes:

With 1 IBM-PC floppy disk (3.5 inch; HD) containing a large general purpose mathematical software collection written in C including a collection of special function classes. The functions are computed for real variables only. (See Lau (2004) for Java version).

External Links:

ISBN 0-8493-7376-X, MathReview, ZentralBlatt

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, H. T. Lau (2004).

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A Numerical Library in Java for Scientists & Engineers. Chapman & Hall/CRC, Boca Raton, FL.

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Notes:

With 1 CD-ROM (Windows, Macintosh and Unix) containing a large general purpose mathematical software collection written in Java including a collection of special function classes. The functions are computed for real variables only. (See Lau (1995) for C version).

External Links:

ISBN 1-58488-430-4, MathReview Entry, ZentralBlatt

Referenced by:

H. T. Lau (1995).

Encodings:

BibTeX

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Eine Verallgemeinerung der Sphäroidfunktionen. Arch. Math. 11, pp. 29–39.

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External Links:

ISSN 0003-9268, Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§30.12.

Encodings:

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The confluent hypergeometric functions $M(a,b;z)$ and $U(a,b;z)$ for large $b$ and $z$ . J. Comput. Appl. Math. 233 (6), pp. 1570–1576.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§13.8(ii), §13.8(ii).

Encodings:

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Large degree asymptotics of generalized Bernoulli and Euler polynomials. J. Math. Anal. Appl. 363 (1), pp. 197–208.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§24.16(i), §24.16(i).

Encodings:

BibTeX

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15: Bibliography O

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Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.

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External Links:

ISSN 1364-5021, Document, MathReview (Donald A. Lutz), ZentralBlatt

Referenced by:

§2.11(v), §2.7(ii).

Encodings:

BibTeX

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Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

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Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

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Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

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External Links:

ISSN 0219-5305, Document, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Tables for Bessel Functions of Moderate or Large Orders. National Physical Laboratory Mathematical Tables, Vol. 6. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.

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External Links:

MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§10.20(i), §10.41(ii), 3rd item, 3rd item.

Encodings:

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16: 3.4 Differentiation

… ►The $B_k^n$ are the differentiated Lagrangian interpolation coefficients: … ►where $c_n$ is defined by (3.3.12), with numerical values as in §3.3(ii). … ►where $C$ is a simple closed contour described in the positive rotational sense such that $C$ and its interior lie in the domain of analyticity of $f$, and $x_0$ is interior to $C$. Taking $C$ to be a circle of radius $r$ centered at $x_0$, we obtain … ►With the choice $r=k$ (which is crucial when $k$ is large because of numerical cancellation) the integrand equals ${\mathrm{e}}^k$ at the dominant points $\theta =0,2\pi$, and in combination with the factor $k^{-k}$ in front of the integral sign this gives a rough approximation to $1/k!$. …

17: Bibliography G

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A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.

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External Links:

FullText, MathReview (D. C. Gilles), ZentralBlatt

Referenced by:

§10.74(v), §3.10(ii).

Encodings:

BibTeX

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Algorithm 819: AIZ, BIZ: two Fortran 77 routines for the computation of complex Airy functions. ACM Trans. Math. Software 28 (3), pp. 325–336.

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Notes:

Double-precision Fortran, maximum accuracy 18S.

External Links:

ISSN 0098-3500, Document, GAMS (module 819; package TOMS), MathReview Entry, ZentralBlatt

Referenced by:

3rd item.

Encodings:

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Algorithm 822: GIZ, HIZ: two Fortran 77 routines for the computation of complex Scorer functions. ACM Trans. Math. Software 28 (4), pp. 436–447.

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Notes:

Double-precision Fortran, maximum accuracy 18S.

External Links:

ISSN 0098-3500, Document, GAMS (module 822; package TOMS), MathReview Entry, ZentralBlatt

Referenced by:

2nd item.

Encodings:

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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:

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GNU Scientific Library The GNU Project.

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Notes:

The GNU Scientific Library is a large general purpose numerical software library with broad coverage of elementary and special functions. Implementation is in double precision.

External Links:

Home Page

Referenced by:

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

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18: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …They tend to thin out among the large integers, but this thinning out is not completely regular. … ►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. … ►Such a set is a reduced residue system modulo $n$. … ►where $p^a$ is a prime power with $a\ge 1$; otherwise $\mathrm{\Lambda }\left(n\right)=0$. …

19: Bibliography P

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Stokes phenomenon demystified. Bull. Inst. Math. Appl. 31 (1-2), pp. 21–28.

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External Links:

ISSN 0905-5628, MathReview Entry, ZentralBlatt

Referenced by:

§2.11(iv).

Encodings:

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Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.

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External Links:

ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§10.20(ii).

Encodings:

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Handbook of the Normal Distribution. Statistics: Textbooks and Monographs, Vol. 40, Marcel Dekker Inc., New York.

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Notes:

A revised and expanded 2nd edition was published in 1996.

External Links:

ISBN 0-8247-1541-1, MathReview (S. Kotz), ZentralBlatt

Referenced by:

§7.20(iii).

Encodings:

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Algorithm 28. Algorithm for the computation of Bessel function integrals. J. Comput. Appl. Math. 11 (1), pp. 119–137.

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External Links:

ISSN 0377-0427, Document, ZentralBlatt

Referenced by:

§10.77(ix).

Encodings:

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A First Course in Real Analysis. 2nd edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York.

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External Links:

ISBN 0-387-97437-7, MathReview (Gerald J. Janusz)

Referenced by:

§1.5(iv), §1.8(i).

Encodings:

BibTeX

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