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11: Bibliography S

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The Laplace Transform: Theory and Applications. Undergraduate Texts in Mathematics, Springer-Verlag, New York.

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External Links:

ISBN 0-387-98698-7, MathReview (Philip Heywood), ZentralBlatt

Referenced by:

§1.14(iii), §1.14(vii).

Encodings:

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The determination of phases of wave functions. Proc. Phys. Soc. 79 (6), pp. 1296–1297.

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External Links:

Document, ZentralBlatt

Referenced by:

§33.23(vii).

Encodings:

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The accuracy of iterated JWBK approximations for Coulomb radial functions. Comput. Phys. Comm. 32 (2), pp. 115–119.

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External Links:

Document

Referenced by:

§33.23(vii).

Encodings:

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Numerical evaluation of the Hankel transform. Comput. Phys. Comm. 116 (2-3), pp. 278–294.

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External Links:

ISSN 0010-4655, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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Some results on the zeros of cylindrical functions and of their derivatives. Rend. Sem. Mat. Univ. Politec. Torino 38 (1), pp. 67–85 (Italian. English summary).

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External Links:

MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

BibTeX

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12: 24.2 Definitions and Generating Functions

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24.2.3 $$\frac{t{\mathrm{e}}^{xt}}{{\mathrm{e}}^t-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_n\left(x\right)\frac{t^n}{n!},$$ .

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $t$: real or complex and $x$: real or complex

A&S Ref:

23.1.1

Referenced by:

§18.2(xii), §24.4(iv), §24.4(vii)

Permalink:

http://dlmf.nist.gov/24.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.2(i), §24.2 and Ch.24

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24.2.8 $$\frac{2{\mathrm{e}}^{xt}}{{\mathrm{e}}^t+1}=\sum _{n=0}^{\mathrm{\infty }}{E}_n\left(x\right)\frac{t^n}{n!},$$ ,

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $t$: real or complex and $x$: real or complex

A&S Ref:

23.1.1

Referenced by:

§18.2(xii), §24.4(iv), §24.4(vii)

Permalink:

http://dlmf.nist.gov/24.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.2(ii), §24.2 and Ch.24

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Table 24.2.5: Coefficients $b_{n,k}$ of the Bernoulli polynomials $B_n\left(x\right)={\sum }_{k=0}^n{b}_{n,k}{x}^k$.

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	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
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$11$	$0$	$\frac{5}{6}$	$0$	$-\frac{11}{2}$	$0$	$11$	$0$	$-11$	$0$	$\frac{55}{6}$	$-\frac{11}{2}$	$1$
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Table 24.2.6: Coefficients $e_{n,k}$ of the Euler polynomials $E_n\left(x\right)={\sum }_{k=0}^n{e}_{n,k}{x}^k$.

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	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
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13: Bibliography H

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Fast Hankel transform algorithm. IEEE Trans. Acoust. Speech Signal Process. 32 (3), pp. 666–671.

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External Links:

ISSN 0096-3518, FullText, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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A Course of Pure Mathematics. 10th edition, Cambridge University Press.

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Notes:

Numerous reprintings exist, including the Centenary Edition with Foreword by T. W. Körner (Cambridge Univerity Press, 2008).

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.4(ii), §1.4(iii), §1.4(iv), §1.4(v), §1.4(vi), §1.4(vii), §4.4(iii), §4.6(i), §4.6(ii).

Encodings:

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Applied and Computational Complex Analysis. Vol. 1: Power Series—Integration—Conformal Mapping—Location of Zeros. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York.

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Notes:

Reprinted in 1988.

External Links:

ISBN 0-471-37244-7, MathReview (M. Marden), ZentralBlatt

Referenced by:

§1.10(vii), §1.9(vi), Ch.3.

Encodings:

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Coulomb Wave Functions. In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.), pp. 408–465.

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External Links:

MathReview (G. Källén)

Referenced by:

§33.2(iii), §33.23(vii), §33.23(vii), §33.24, §33.5(ii), §33.7, Ch.33.

Encodings:

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Analytical structure and properties of Coulomb wave functions for real and complex energies. Ann. Physics 155 (2), pp. 461–493.

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External Links:

ISSN 0003-4916, Document, MathReview

Referenced by:

§33.10(ii), §33.13, §33.22(vii).

Encodings:

BibTeX

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14: Bibliography B

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The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

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External Links:

ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt

Referenced by:

§18.18(vii).

Encodings:

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Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. Appl. Math. Lett. 9 (5), pp. 21–26.

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External Links:

ISSN 0893-9659, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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KLEIN: Coulomb functions for real $\lambda$ and positive energy to high accuracy. Comput. Phys. Comm. 24 (2), pp. 141–159.

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Notes:

Double-precision Fortran code for positive energies.

External Links:

Document, Code

Referenced by:

5th item, §33.23(vii), 1st item.

Encodings:

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On numerical evaluation of integrals involving Bessel functions. Apl. Mat. 31 (5), pp. 396–410.

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External Links:

ISSN 0373-6725, MathReview (G. A. Evans), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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The determination of phases and amplitudes of wave functions. Proc. Phys. Soc. 81 (3), pp. 442–452.

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External Links:

Document

Referenced by:

§33.23(vii).

Encodings:

BibTeX

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15: Bibliography K

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Bernstein, Pick, Poisson and related integral expressions for Lambert $W$ . Integral Transforms Spec. Funct. 23 (11), pp. 817–829.

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External Links:

ISSN 1065-2469, Document, Link, MathReview (Ross C. McPhedran), ZentralBlatt

Referenced by:

§4.13.

Encodings:

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Differentialgleichungen: Lösungsmethoden und Lösungen. Teil I. B. G. Teubner, Stuttgart (German).

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External Links:

ISBN 3-519-02017-3, MathReview, ZentralBlatt

Referenced by:

§1.13(vii), §10.13, §4.20, §4.34, §4.7(ii).

Encodings:

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Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§19.12.

Encodings:

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Introduction to Solid State Physics. 7th Edition edition, John Wiley and Sons, New York.

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External Links:

ISBN 0-471-11181-3

Referenced by:

§1.18(vii).

Encodings:

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Algorithm 327: Dilogarithm [S22]. Comm. ACM 11 (4), pp. 270–271.

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Notes:

Includes Algol program, maximum accuracy 12D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§25.21(v).

Encodings:

BibTeX

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16: Bibliography M

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Asymptotic expansions for the zeros of certain special functions. J. Comput. Appl. Math. 145 (2), pp. 261–267.

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External Links:

ISSN 0377-0427, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.70, §11.4(vii), §6.13.

Encodings:

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On the $G$-function. VII, VIII. Nederl. Akad. Wetensch., Proc. 49, pp. 1063–1072, 1165–1175 = Indagationes Math. 8, 661–670, 713–723 (1946).

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§16.11(ii).

Encodings:

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An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.

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External Links:

ISBN 0-471-58884-9, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii).

Encodings:

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Jacobian Elliptic Function Tables. Dover Publications Inc., New York.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§22.20(vii).

Encodings:

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Stirling numbers of the second kind. Duke Math. J. 25 (1), pp. 29–43.

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External Links:

Document, MathReview (A. Sade), ZentralBlatt

Referenced by:

§26.1, §26.8(vii), Notations S.

Encodings:

BibTeX

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17: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►These cases are treated in §§12.10(vii)–12.10(viii). … ►

12.10.23 $$\eta =\frac{1}{2}\arccos t-\frac{1}{2}t\sqrt{1-t^2},$$

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Defines:

$\eta$ (locally)

Symbols:

$\arccos z$: arccosine function

Referenced by:

§12.10(vii), §12.14(ix)

Permalink:

http://dlmf.nist.gov/12.10.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(iv), §12.10 and Ch.12

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§12.10(vii) Negative $a$, . Expansions in Terms of Airy Functions

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12.10.40 $$\varphi (\zeta )={\left(\frac{\zeta }{t^2-1}\right)}^{\frac{1}{4}}.$$

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Defines:

$\varphi (\zeta )$: function (locally)

Symbols:

$\zeta$: change of variable

Referenced by:

§12.10(vii)

Permalink:

http://dlmf.nist.gov/12.10.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vii), §12.10 and Ch.12

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12.10.41 $$t=1+w-\frac{1}{10}{w}^2+\frac{11}{350}{w}^3-\frac{823}{63000}{w}^4+\frac{\mathrm{1\hspace{0.33em}50653}}{\mathrm{242\hspace{0.33em}55000}}{w}^5+\mathrm{\cdots },$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $\zeta$: change of variable

Referenced by:

§12.11(iii), §12.11(iii)

Permalink:

http://dlmf.nist.gov/12.10.E41

Encodings:

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See also:

Annotations for §12.10(vii), §12.10 and Ch.12

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18: 25.21 Software

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§25.21(vii) Fermi–Dirac and Bose–Einstein Integrals

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19: 9.18 Tables

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National Bureau of Standards (1958) tabulates $A_0(x)\equiv \pi \mathrm{Hi}\left(-x\right)$ and $-A_0^{\prime }(x)\equiv \pi {\mathrm{Hi}}^{\prime }\left(-x\right)$ for $x=0(.01)1(.02)5(.05)11$ and $1/x=0.01(.01)0.1$; ${\int }_0^x{A}_0(t)dt$ for $x=0.5,1(1)11$. Precision is 8D.

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Gil et al. (2003c) tabulates the only positive zero of ${\mathrm{Gi}}^{\prime }\left(z\right)$, the first 10 negative real zeros of $\mathrm{Gi}\left(z\right)$ and ${\mathrm{Gi}}^{\prime }\left(z\right)$, and the first 10 complex zeros of $\mathrm{Gi}\left(z\right)$, ${\mathrm{Gi}}^{\prime }\left(z\right)$, $\mathrm{Hi}\left(z\right)$, and ${\mathrm{Hi}}^{\prime }\left(z\right)$. Precision is 11 or 12S.

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§9.18(vii) Generalized Airy Functions

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20: 3.6 Linear Difference Equations

… ►In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome. … ►

Table 3.6.1: Weber function $w_n={𝐄}_n\left(1\right)$ computed by Olver’s algorithm.

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$n$	$p_n$		$e_n$		$e_n/(p_n{p}_{n+1})$		$w_n$
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11	$\mathrm{0.29154\hspace{0.33em}738}$	$\mathrm{\times 10¹⁰}$	$\mathrm{0.37225\hspace{0.33em}201}$	$\mathrm{\times 10¹⁰}$	$\mathrm{0.19952\hspace{0.33em}026}$	$\mathrm{\times 10⁻¹⁰}$	$\mathrm{0.58373\hspace{0.33em}946}$	$\mathrm{\times 10⁻¹}$
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§3.6(vii) Linear Difference Equations of Other Orders

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3.6.17 $$a_n{w}_{n+1}-b_n{w}_n=d_n.$$

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Symbols:

$w_n$: sequence, $a_n$: coefficient, $b_n$: coefficient and $d_n$: coefficient

Permalink:

http://dlmf.nist.gov/3.6.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §3.6(vii), §3.6 and Ch.3

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3.6.18 $$a_{n,k}{w}_{n+k}+a_{n,k-1}{w}_{n+k-1}+\mathrm{\cdots }+a_{n,0}{w}_n=d_n,$$

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Symbols:

$w_n$: sequence, $a_n$: coefficient and $d_n$: coefficient

Permalink:

http://dlmf.nist.gov/3.6.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §3.6(vii), §3.6 and Ch.3

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