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11: 19.24 Inequalities

… ►The condition $y\le z$ for (19.24.1) and (19.24.2) serves only to identify $y$ as the smaller of the two nonzero variables of a symmetric function; it does not restrict validity. … ►If $y$, $z$, and $p$ are positive, then … ►All variables are positive, and equality occurs iff all variables are equal. … ►If $a$ ($\ne 0$) is real, all components of $𝐛$ and $𝐳$ are positive, and the components of $z$ are not all equal, then …

12: 10.23 Sums

… ►If $𝒞=J$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary. … ►The restriction is unnecessary when $𝒞=J$ and $\nu$ is an integer. … ►If $u$, $v$ are real and positive and $0\le \alpha \le \pi$, then $w$ and $\chi$ are real and nonnegative, and the geometrical relationship is shown in Figure 10.23.1. … ►The restriction is unnecessary in (10.23.7) when $𝒞=J$ and $\nu$ is an integer, and in (10.23.8) when $𝒞=J$. …

13: Bibliography B

… ►

Random and Restricted Walks: Theory and Applications. Gordon and Breach, New York.

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External Links:

ZentralBlatt

Referenced by:

§16.24(i).

Encodings:

BibTeX

… ►

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:

BibTeX

►

COULFG: Coulomb and Bessel functions and their derivatives, for real arguments, by Steed’s method. Comput. Phys. Comm. 27, pp. 147–166.

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Notes:

Double-precision Fortran code for positive energies. Maximum accuracy: 31S.

External Links:

Document, Code

Referenced by:

2nd item, 2nd item.

Encodings:

BibTeX

►

The Calculation of Spherical Bessel Functions and Coulomb Functions. In Computational Atomic Physics: Electron and Positron Collisions with Atoms and Ions, K. Bartschat and J. Hinze (Eds.), pp. 181–202.

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Notes:

Includes program disk. Double-precision Fortran code for positive energies. Maximum accuracy: 15D.

External Links:

ISBN 3-540-60179-1, FullText and Code

Referenced by:

3rd item, §33.8.

Encodings:

BibTeX

… ►

Bessel Functions. Part II: Functions of Positive Integer Order. British Association for the Advancement of Science, Mathematical Tables, Volume 10, Cambridge University Press, Cambridge.

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Notes:

Reprinted, with corrections, in 1960.

External Links:

ISBN 0-521-04321-2, MathReview (R. C. Archibald), ZentralBlatt

Referenced by:

§10.18(iii), §10.19(ii), §10.21(vi), §10.41(ii), 2nd item, 2nd item, §3.6(iii).

Encodings:

BibTeX

…

14: Guide to Searching the DLMF

… ►From there you can also access an advanced search page where you can control certain settings, narrowing the search to certain chapters, or restricting the results to equations, graphs, tables, or bibliographic items. … ►

adj, prec/n, and near/n, where n is any positive natural number.

…

15: 18.15 Asymptotic Approximations

… ►Here, and elsewhere in §18.15, $\delta$ is an arbitrary small positive constant. … ►with $c$ denoting an arbitrary positive constant. … ►Here $J_{\nu }\left(z\right)$ denotes the Bessel function (§10.2(ii)), $\mathrm{env}{J}_{\nu }\left(z\right)$ denotes its envelope (§2.8(iv)), and $\delta$ is again an arbitrary small positive constant. … ►For more powerful asymptotic expansions as $n\to \mathrm{\infty }$ in terms of elementary functions that apply uniformly when , $-1+\delta \le t\le 1-\delta$, or , where $t=x/\sqrt{2n+1}$ and $\delta$ is again an arbitrary small positive constant, see §§12.10(i)–12.10(iv) and 12.10(vi). … ►These approximations apply when the parameters are large, namely $\alpha$ and $\beta$ (subject to restrictions) in the case of Jacobi polynomials, $\lambda$ in the case of ultraspherical polynomials, and $|\alpha |+|x|$ in the case of Laguerre polynomials. …

16: 8.21 Generalized Sine and Cosine Integrals

…

17: 22.19 Physical Applications

… ►Also, $\theta (t)$ is not restricted to the principal range $-\pi \le \theta \le \pi$. … ►

22.19.5 $$V(x)=\pm \frac{1}{2}{x}^2\pm \frac{1}{4}\beta x^4$$

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Symbols:

$V(x)$: potential energy, $x(t)$: coordinate and $\beta$: real positive

Permalink:

http://dlmf.nist.gov/22.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.19(ii), §22.19 and Ch.22

… ►For $\beta$ real and positive, three of the four possible combinations of signs give rise to bounded oscillatory motions. …The subsequent position as a function of time, $x(t)$, for the three cases is given with results expressed in terms of $a$ and the dimensionless parameter $\eta =\frac{1}{2}\beta a^2$. …

18: 5.11 Asymptotic Expansions

… ►

5.11.8 $$\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)\sim \left(z+h-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=2}^{\mathrm{\infty }}\frac{{(-1)}^k{B}_k\left(h\right)}{k(k-1)z^{k-1}},$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§12.10(ii), (5.11.8), §5.11(i), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8), Erratum (V1.0.11) for Equation (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E8

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.11):

Originally $h$ in (5.11.8) was unnecessarily restricted to lie in the interval $[0,1]$. In fact, $h$ may lie anywhere in the complex plane.

Suggested 2015-02-28 by Nico Temme

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z+h\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

… ►If the sums in the expansions (5.11.1) and (5.11.2) are terminated at $k=n-1$ ($k\ge 0$) and $z$ is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. …

19: 31.3 Basic Solutions

… ►If the other exponent is not a positive integer, that is, if $\gamma \ne 0,-1,-2,\mathrm{\dots }$, then from §2.7(i) it follows that $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ exists, is analytic in the disk , and has the Maclaurin expansion … ►With similar restrictions to those given in §31.3(i), the following results apply. …

20: 18.36 Miscellaneous Polynomials

… ►Orthogonality of the the classical OP’s with respect to a positive weight function, as in Table 18.3.1 requires, via Favard’s theorem, $A_n{A}_{n-1}{C}_n>0$ for $n\ge 1$ as per (18.2.9_5). For the Laguerre polynomials $L_n^{(\alpha )}\left(x\right)$ this requires, omitting all strictly positive factors, … ►Exceptional type I $X_m$-EOP’s, form a complete orthonormal set with respect to a positive measure, but the lowest order polynomial in the set is of order $m$, or, said another way, the first $m$ polynomial orders, $0,1,\mathrm{\dots },m-1$ are missing. … ►The restriction to $n\ge 1$ is now apparent: (18.36.7) does not posses a solution if $y(x)$ is a constant. …