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21: 22.19 Physical Applications

… ► $\theta$ being the angular displacement from the point of stable equilibrium, $\theta =0$. … ►Many nonlinear ordinary and partial differential equations have solutions that may be expressed in terms of Jacobian elliptic functions. These include the time dependent, and time independent, nonlinear Schrödinger equations (NLSE) (Drazin and Johnson (1993, Chapter 2), Ablowitz and Clarkson (1991, pp. 42, 99)), the Korteweg–de Vries (KdV) equation (Kruskal (1974), Li and Olver (2000)), the sine-Gordon equation, and others; see Drazin and Johnson (1993, Chapter 2) for an overview. … ►The classical rotation of rigid bodies in free space or about a fixed point may be described in terms of elliptic, or hyperelliptic, functions if the motion is integrable (Audin (1999, Chapter 1)). …A more abstract overview is Audin (1999, Chapters III and IV), and a complete discussion of analytical solutions in the elliptic and hyperelliptic cases appears in Golubev (1960, Chapters V and VII), the original hyperelliptic investigation being due to Kowalevski (1889). …

22: 10.9 Integral Representations

… ►

10.9.30 $$J_{\nu }^2\left(z\right)+Y_{\nu }^2\left(z\right)=\frac{8}{{\pi }^2}{\int }_0^{\mathrm{\infty }}\cosh \left(2\nu t\right)K_0\left(2z\sinh t\right)dt,$$ .

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.9.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(iii), §10.9(iii), §10.9 and Ch.10

… ►For collections of integral representations of Bessel and Hankel functions see Erdélyi et al. (1953b, §§7.3 and 7.12), Erdélyi et al. (1954a, pp. 43–48, 51–60, 99–105, 108–115, 123–124, 272–276, and 356–357), Gröbner and Hofreiter (1950, pp. 189–192), Marichev (1983, pp. 191–192 and 196–210), Magnus et al. (1966, §3.6), and Watson (1944, Chapter 6).

23: Bibliography H

… ►

The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:

MathReview (L. Berg)

Referenced by:

§7.14(iii).

Encodings:

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… ►

Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

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Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

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Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:

MathReview (M. Lawrence Glasser)

Referenced by:

§7.14(iii).

Encodings:

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… ►

Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.

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External Links:

ZentralBlatt

Referenced by:

§7.18(iii), §7.18(iv), §7.18(v), §7.18(vi).

Encodings:

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…

24: 18.39 Applications in the Physical Sciences

… ►While non-normalizable continuum, or scattering, states are mentioned, with appropriate references in what follows, focus is on the $L^2$ eigenfunctions corresponding to the point, or discrete, spectrum, and representing bound rather than scattering states, these former being expressed in terms of OP’s or EOP’s. … ►The properties of $V(x)$ determine whether the spectrum, this being the set of eigenvalues of $\mathscr{H}$, is discrete, continuous, or mixed, see §1.18. … ►See Titchmarsh (1962a, pp. 99–100). … ►From (18.9.23) and (18.5.5) with Table 18.5.1, line 8: … ►Full expressions for both $A_{x_i,l}$ and $B_l(x)$ are given in Yamani and Reinhardt (1975) and it is seen that ${|A_{x_i,l}/B_l(x_i)|}^2$ = $w_i^N/w^{\mathrm{CP}}(x_i)$ where $w_i^N$ is the Gaussian-Pollaczek quadrature weight at $x=x_i$, and $w^{\mathrm{CP}}(x_i)$ is the Gaussian-Pollaczek weight function at the same quadrature abscissa. …

25: Bibliography D

… ►

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).

Encodings:

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Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).

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External Links:

MathReview (I. I. Hirschman), ZentralBlatt

Referenced by:

§2.5(i).

Encodings:

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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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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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External Links:

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

Referenced by:

§2.11(iii), §8.20(ii).

Encodings:

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Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.

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External Links:

ISSN 0021-9045, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.24.

Encodings:

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26: 26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

► $D(m,n)$ is the number of paths from $(0,0)$ to $(m,n)$ that are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. ►

26.6.1 $$D(m,n)=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)\left(\genfrac{}{}{0.0pt}{}{m+n-k}{n}\right)=\sum _{k=0}^n{2}^k\left(\genfrac{}{}{0.0pt}{}{m}{k}\right)\left(\genfrac{}{}{0.0pt}{}{n}{k}\right).$$

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Defines:

$D(m,n)$: Delannoy number (locally)

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.6.E1

Encodings:

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See also:

Annotations for §26.6(i), §26.6(i), §26.6 and Ch.26

… ►

Table 26.6.1: Delannoy numbers $D(m,n)$.

► ►►

$m$	$n$
	…

►

… ►

26.6.4 $$r(n)=D(n,n)-D(n+1,n-1),$$ $n\ge 1$.

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Defines:

$r(n)$: Schröder number (locally)

Symbols:

$n$: nonnegative integer and $D(m,n)$: Delannoy number

Referenced by:

§26.6(i)

Permalink:

http://dlmf.nist.gov/26.6.E4

Encodings:

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See also:

Annotations for §26.6(i), §26.6(i), §26.6 and Ch.26

…

27: Bibliography B

… ►

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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On the power function of the likelihood ratio test for MANOVA. J. Multivariate Anal. 82 (2), pp. 416–421.

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External Links:

ISSN 0047-259X, Document, MathReview (Yasuko Chikuse), ZentralBlatt

Referenced by:

§35.9.

Encodings:

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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.

Encodings:

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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).

Encodings:

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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).

Encodings:

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28: 12.14 The Function $W(a,x)$

… ►For the modulus functions $\tilde{F}(a,x)$ and $\tilde{G}(a,x)$ see §12.14(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). …

29: Bibliography P

… ►

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).

Encodings:

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Unbiasedness of invariant tests for MANOVA and other multivariate problems. Ann. Statist. 8 (6), pp. 1326–1341.

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External Links:

ISSN 0090-5364, FullText, MathReview (Takeaki Kariya), ZentralBlatt

Referenced by:

§35.9.

Encodings:

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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).

Encodings:

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Primzahlverteilung. Die Grundlehren der mathematischen Wissenschaften, Vol. 91, Springer-Verlag, Berlin-Göttingen-Heidelberg (German).

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Notes:

Reprinted in 1978.

External Links:

MathReview (A. E. Ingham), ZentralBlatt

Referenced by:

§27.11.

Encodings:

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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).

Encodings:

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30: 8.23 Statistical Applications

… ►In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of $Q(a,x)$; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …