separation constant

SearchAdvancedHelp

(0.002 seconds)

31—35 of 35 matching pages

31: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Note that the integral in (1.18.66) is not singular if approached separately from above, or below, the real axis: in fact analytic continuation from the upper half of the complex plane, across the cut, and onto higher Riemann Sheets can access complex poles with singularities at discrete energies ${\lambda }_{\mathrm{res}}-\mathrm{i}{\mathrm{\Gamma }}_{\mathrm{res}}/2$ corresponding to quantum resonances, or decaying quantum states with lifetimes proportional to $1/{\mathrm{\Gamma }}_{\mathrm{res}}$. …

32: Bibliography B

… ►

Constant mean curvature surfaces and integrable equations. Uspekhi Mat. Nauk 46 (4(280)), pp. 3–42, 192 (Russian).

ⓘ

Notes:

In Russian. English translation: Russian Math. Surveys, 46(1991), no. 4, pp. 1–45

External Links:

ISSN 0042-1316, MathReview (P. E. Markov), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

… ►

Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§31.13.

Encodings:

BibTeX

…

33: Bibliography P

… ►

Tables of the Incomplete $\mathrm{\Gamma }$-function. Biometrika Office, Cambridge University Press, Cambridge.

ⓘ

Referenced by:

3rd item.

Encodings:

BibTeX

… ►

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

ⓘ

Notes:

With a separately available computer disk

External Links:

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

Referenced by:

§16.23(iv).

Encodings:

BibTeX

…

34: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

… ►

17.6.28 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{\alpha },q^{\beta }}{q^{\gamma }};q,z)=\frac{{\mathrm{\Gamma }}_q\left(\gamma \right)}{{\mathrm{\Gamma }}_q\left(\beta \right){\mathrm{\Gamma }}_q\left(\gamma -\beta \right)}{\int }_0^1\frac{t^{\beta -1}{(tq;q)}_{\gamma -\beta -1}}{{(xt;q)}_{\alpha }}{d}_{q}t.$$

ⓘ

Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $z$: complex variable and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.6.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §17.6(v), §17.6 and Ch.17

… ►where , , and the contour of integration separates the poles of ${(q^{1+\zeta },c{q}^{\zeta };q)}_{\mathrm{\infty }}/\sin \left(\pi \zeta \right)$ from those of $1/{(a{q}^{\zeta },b{q}^{\zeta };q)}_{\mathrm{\infty }}$, and the infimum of the distances of the poles from the contour is positive. …

35: 2.1 Definitions and Elementary Properties

… ►(Here and elsewhere in this chapter $\delta$ is an arbitrary small positive constant.) … ►For example, suppose $f(x)$ is continuous and $f(x)\sim x^{\nu }$ as $x\to +\mathrm{\infty }$ in $ℝ$, where $\nu$ ($\in ℂ$) is a constant. Then … ►

2.1.12

ⓘ

Symbols:

$\sim$: asymptotic equality, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $f(x)$: continuous function and $\nu$: complex constant

Permalink:

http://dlmf.nist.gov/2.1.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.1(ii), §2.1 and Ch.2

… ►In those cases it is usually necessary to interpret each infinite series separately in the manner described above; that is, it is not always possible to reinterpret the asymptotic approximation as a single asymptotic expansion. …