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21: Bibliography L

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Exact computation of the $9$-$j$ symbols. Comput. Phys. Comm. 70 (3), pp. 544–556.

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Notes:

Double-precision Fortran

External Links:

ISSN 0010-4655, Document, Code, MathReview

Referenced by:

7th item.

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Elliptic Functions and Applications. Applied Mathematical Sciences, Vol. 80, Springer-Verlag, New York.

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External Links:

ISBN 0-387-96965-9, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.2(ii), §19.36(iii), §20.13, §20.15, §20.2(i), §20.2(ii), §20.2(iii), §20.2(iv), §20.4(i), §20.5(i), §20.5(ii), §20.7(i), §20.7(ii), §20.7(iv), §20.7(vi), §20.7(vii), §20.7(vii), §20.7(viii), Ch.20, §22.10(i), §22.10(ii), §22.12, §22.13(i), §22.14(i), §22.14(ii), §22.14(iii), §22.14(iii), §22.15(i), §22.15(ii), §22.15(ii), §22.16(ii), §22.16(iii), §22.17(i), §22.18(i), §22.18(iii), §22.19(ii), §22.19(i), §22.19(i), §22.19(ii), §22.19(ii), §22.19(iv), §22.19(v), §22.2, §22.20(vii), §22.21, §22.4(i), §22.4(iii), §22.5(i), §22.5(ii), §22.6(ii), §22.6(iii), §22.6(iv), §22.7(i), §22.7(ii), §22.8(i), §22.8(ii), Ch.22, §23.10(i), §23.10(i), §23.10(iii), §23.10(iv), §23.12, §23.2(ii), §23.2(iii), §23.2(iii), §23.21(i), §23.3(i), §23.3(ii), §23.6(iv), §23.6(i), §23.6(i), §23.6(ii), §23.6(ii), §23.6(iii), §23.7, §23.8(i), §23.8(ii), §23.8(iii), Ch.23.

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On the asymptotics of the Meixner-Pollaczek polynomials and their zeros. Constr. Approx. 17 (1), pp. 59–90.

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ISSN 0176-4276, Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§18.24.

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Statistical Physics, Part 2: Theory of the Condensed State. Pergamon Press, Oxford.

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Notes:

Course of Theoretical Physics , Vol. 9. Translated from the Russian by J. B. Sykes and M. J. Kearsley.

External Links:

ISBN 0-08-023073-3; 0-08-023072-5, MathReview

Referenced by:

§25.17.

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Deep water ship-waves. Phil. Mag. 9, pp. 733–757.

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External Links:

ZentralBlatt

Referenced by:

§36.13.

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BibTeX

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22: Bibliography B

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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).

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Avoided crossings of the quartic oscillator. J. Phys. A 30 (9), pp. 3057–3067.

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External Links:

ISSN 0305-4470, Document, MathReview (Gabriel Alvarez), ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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Methods of calculation of radial wave functions and new tables of Coulomb functions. Physical Rev. (2) 80, pp. 553–560.

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External Links:

Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§33.7.

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From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. London Math. Soc. (3) 90 (1), pp. 167–208.

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External Links:

ISSN 0024-6115, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.9(iii).

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Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7 (1), pp. 78–90.

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Notes:

Algol 60 programs are included for real and complex elliptic integrals, and for Jacobian elliptic functions of real argument.

External Links:

ISSN 0029-599X, Document, MathReview (M. Feliciano), ZentralBlatt

Referenced by:

§19.1, §19.2(iii), §19.36(ii), §19.39(iii), §22.22(ii).

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23: 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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Yakovleva (1969) tabulates Fock’s functions $U(x)\equiv \sqrt{\pi }\mathrm{Bi}\left(x\right)$, $U^{\prime }(x)=\sqrt{\pi }{\mathrm{Bi}}^{\prime }\left(x\right)$, $V(x)\equiv \sqrt{\pi }\mathrm{Ai}\left(x\right)$, $V^{\prime }(x)=\sqrt{\pi }{\mathrm{Ai}}^{\prime }\left(x\right)$ for $x=-9(.001)9$. Precision is 7S.

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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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Nosova and Tumarkin (1965) tabulates $e_0(x)\equiv \pi \mathrm{Hi}\left(-x\right)$, $e_0^{\prime }(x)=-\pi {\mathrm{Hi}}^{\prime }\left(-x\right)$, ${\tilde{e}}_0(-x)\equiv -\pi \mathrm{Gi}\left(x\right)$, ${\tilde{e}}_0^{\prime }(-x)=\pi {\mathrm{Gi}}^{\prime }\left(x\right)$ for $x=-1(.01)10$; 7D. Also included are the real and imaginary parts of $e_0(z)$ and $\mathrm{i}{e}_0^{\prime }(z)$, where $z=\mathrm{i}y$ and $y=0(.01)9$; 6-7D.

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24: 12.14 The Function $W(a,x)$

… ►Other expansions, involving $\cos \left(\frac{1}{4}{x}^2\right)$ and $\sin \left(\frac{1}{4}{x}^2\right)$, can be obtained from (12.4.3) to (12.4.6) by replacing $a$ by $-\mathrm{i}a$ and $z$ by $x{\mathrm{e}}^{\pi \mathrm{i}/4}$; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). … ►Here ${𝒜}_s(t)$ is as in §12.10(ii), $\sigma$ is defined by … ►uniformly for $t\in [-1+\delta ,1-\delta ]$, with $\eta$ given by (12.10.23) and ${\widetilde{𝒜}}_s(t)$ given by (12.10.24). … ►uniformly for $t\in [-1+\delta ,\mathrm{\infty })$, with $\zeta$, $\varphi (\zeta )$, $A_s(\zeta )$, and $B_s(\zeta )$ as in §12.10(vii). … ►For properties of the modulus and phase functions, including differential equations and asymptotic expansions for large $x$, see Miller (1955, pp. 87–88). …

25: Bibliography D

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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.

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External Links:

ISSN 0022-314X, Document, MathReview (T. Metsänkylä)

Referenced by:

§24.12(iii).

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Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).

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Notes:

With an appendix by I. M. Krichever. In Russian. English translation: Russian Math. Surveys 36(1981), no. 2, pp. 11–92 (1982)

External Links:

ISSN 0042-1316, Document, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§21.1, §21.6(ii), §21.7(i), Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2, Ch.21, Notations T.

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Lamé instantons. J. High Energy Phys. 2000 (1), pp. Paper 19, 8.

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Notes:

(electronic only, article id. JHEP01(2000)019, 8 pp.).

External Links:

ISSN 1029-8479, Document, MathReview, ZentralBlatt

Referenced by:

§22.11.

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Error analysis in a uniform asymptotic expansion for the generalised exponential integral. J. Comput. Appl. Math. 80 (1), pp. 127–161.

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ISSN 0377-0427, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§2.11(iii), §8.20(ii).

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Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.

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External Links:

Document, MathReview (R. C. Varma), ZentralBlatt

Referenced by:

§18.17(iii).

Encodings:

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26: 22.21 Tables

… ►Spenceley and Spenceley (1947) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$, $\mathrm{am}(Kx,k)$, $ℰ(Kx,k)$ for $\arcsin k=1^{\circ }(1^{\circ }){89}^{\circ }$ and $x=0\left(\frac{1}{90}\right)1$ to 12D, or 12 decimals of a radian in the case of $\mathrm{am}(Kx,k)$. … ►Lawden (1989, pp. 280–284 and 293–297) tabulates $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, $\mathrm{dn}(x,k)$, $ℰ(x,k)$, $\mathrm{Z}\left(x|k\right)$ to 5D for $k=0.1(.1)0.9$, $x=0(.1)X$, where $X$ ranges from 1. …

27: Bibliography P

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A new basis for the representation of the rotation group. Lamé and Heun polynomials. J. Mathematical Phys. 14 (8), pp. 1130–1139.

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External Links:

Document, ZentralBlatt

Referenced by:

§29.18(iv).

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$A=B$ . A K Peters Ltd., Wellesley, MA.

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Notes:

With a separately available computer disk

External Links:

ISBN 1-56881-063-6, MathReview (Peter Paule), ZentralBlatt

Referenced by:

§16.23(iv).

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Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.37.

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On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9 (4), pp. 404–407.

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External Links:

Document, MathReview, ZentralBlatt

Referenced by:

§3.11(ii).

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Stacking models of vesicles and compact clusters. J. Statist. Phys. 80 (3–4), pp. 755–779.

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External Links:

Document, ZentralBlatt

Referenced by:

§8.24(ii).

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28: 36.5 Stokes Sets

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36.5.4 $$80x^5-40x^4-55x^3+5x^2+20x-1=0,$$

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Symbols:

$x$: real parameter

Permalink:

http://dlmf.nist.gov/36.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.11 $$\frac{x}{z^2}=-1-12u^2+8u-\left|\frac{y}{z^2}\right|\frac{\frac{1}{3}-u}{{\left(u\left(\frac{2}{3}-u\right)\right)}^{1/2}}.$$

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Symbols:

$y$: real parameter, $z$: real parameter and $x$: real parameter

Referenced by:

§36.5(iv)

Permalink:

http://dlmf.nist.gov/36.5.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

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36.5.12 $$8u^3-4u^2-\left|\frac{y}{3z^2}\right|{\left(\frac{u}{\frac{2}{3}-u}\right)}^{1/2}=\frac{y^2}{6w{z}^4}-2w^3-2w^2,$$

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Symbols:

$y$: real parameter, $z$: real parameter and $w$: quantity

Permalink:

http://dlmf.nist.gov/36.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

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$Y(u,X)=8u-24u^2+X{\displaystyle \frac{u-\frac{1}{6}}{{\left(u\left(u-\frac{1}{3}\right)\right)}^{1/2}}},$

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29: Bibliography E

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A formula including Legendre’s $E{K}^{\prime }+K{E}^{\prime }-K{K}^{\prime }=\frac{1}{2}\pi$ . Messenger of Math. 33, pp. 31–32.

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Notes:

Errata on p. 45 of the same volume.

External Links:

ZentralBlatt

Referenced by:

§15.16.

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The Fuchsian equation of second order with four singularities. Duke Math. J. 9 (1), pp. 48–58.

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External Links:

Document, MathReview (R. E. Langer), ZentralBlatt

Referenced by:

§31.11(i).

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Algorithm 934: Fortran 90 subroutines to compute Mathieu functions for complex values of the parameter. ACM Trans. Math. Softw. 40 (1), pp. 8:1–8:19.

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External Links:

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

Referenced by:

4th item, §28.36(iii).

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Algorithm 861: Fortran 90 subroutines for computing the expansion coefficients of Mathieu functions using Blanch’s algorithm. ACM Trans. Math. Software 32 (4), pp. 622–634.

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External Links:

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

Referenced by:

3rd item, §28.36(iii).

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On the representations of a number as a sum of three squares. Proc. London Math. Soc. (3) 9, pp. 575–594.

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External Links:

Document, MathReview (S. Chowla), ZentralBlatt

Referenced by:

§20.12(i).

Encodings:

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30: 28.35 Tables

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Blanch and Clemm (1965) includes values of ${\mathrm{Mc}}_n^{(2)}(x,\sqrt{q})$, $\mathrm{Mc}_n^{(2)}{}_{}{}^{\prime }(x,\sqrt{q})$ for $n=0(1)7$, $x=0(.02)1$; $n=8(1)15$, $x=0(.01)1$. Also ${\mathrm{Ms}}_n^{(2)}(x,\sqrt{q})$, $\mathrm{Ms}_n^{(2)}{}_{}{}^{\prime }(x,\sqrt{q})$ for $n=1(1)7$, $x=0(.02)1$; $n=8(1)15$, $x=0(.01)1$. In all cases $q=0(.05)1$. Precision is generally 7D. Approximate formulas and graphs are also included.

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Ince (1932) includes eigenvalues $a_n$, $b_n$, and Fourier coefficients for $n=0$ or $1(1)6$, $q=0(1)10(2)20(4)40$; 7D. Also ${\mathrm{ce}}_n(x,q)$, ${\mathrm{se}}_n(x,q)$ for $q=0(1)10$, $x=1(1)90$, corresponding to the eigenvalues in the tables; 5D. Notation: $a_n={\mathrm{𝑏𝑒}}_n-2q$, $b_n={\mathrm{𝑏𝑜}}_n-2q$.

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Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $a_n\left(q\right)$, $b_{n+1}\left(q\right)$ for $n=0(1)4$, $q=0(1)50$; $n=0(1)20$ ($a$’s) or 19 ($b$’s), $q=1,3,5,10,15,25,50(50)200$. Fourier coefficients for ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_{n+1}(x,10)$, $n=0(1)7$. Mathieu functions ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_{n+1}(x,10)$, and their first $x$-derivatives for $n=0(1)4$, $x=0(5^{\circ }){90}^{\circ }$. Modified Mathieu functions ${\mathrm{Mc}}_n^{(j)}(x,\sqrt{10})$, ${\mathrm{Ms}}_{n+1}^{(j)}(x,\sqrt{10})$, and their first $x$-derivatives for $n=0(1)4$, $j=1,2$, $x=0(.2)4$. Precision is mostly 9S.

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Blanch and Clemm (1969) includes eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$ for $q=\rho {\mathrm{e}}^{\mathrm{i}\varphi }$, $\rho =0(.5)25$, $\varphi =5^{\circ }(5^{\circ }){90}^{\circ }$, $n=0(1)15$; 4D. Also $a_n\left(q\right)$ and $b_n\left(q\right)$ for $q=\mathrm{i}\rho$, $\rho =0(.5)100$, $n=0(2)14$ and $n=2(2)16$, respectively; 8D. Double points for $n=0(1)15$; 8D. Graphs are included.

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Zhang and Jin (1996, pp. 533–535) includes the zeros (in degrees) of ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_n(x,10)$ for $n=1(1)10$, and the first 5 zeros of ${\mathrm{Mc}}_n^{(j)}(x,\sqrt{10})$, ${\mathrm{Ms}}_n^{(j)}(x,\sqrt{10})$ for $n=0$ or $1(1)8$, $j=1,2$. Precision is mostly 9S.

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