separation constant

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1: 28.32 Mathematical Applications

… ►The separated solutions

V(\xi,\eta)=v(\xi)w(\eta)

can be obtained from the modified Mathieu’s equation (28.20.1) for

v

and from Mathieu’s equation (28.2.1) for

w

, where

a

is the separation constant and

q=\tfrac{1}{4}c^{2}k^{2}

. … ► … ►is separated in this system, each of the separated equations can be reduced to the Whittaker–Hill equation (28.31.1), in which

A, B

are separation constants. …

2: 31.10 Integral Equations and Representations

… ►

Kernel Functions

… ►where

\sigma

is a separation constant. … ►

31.10.19

\mathcal{K}(u,v,w)=u^{1-\gamma}v^{1-\delta}w^{1-\epsilon}\mathscr{C}_{1-\gamma% }\left(u\sqrt{\sigma_{1}}\right)\*\mathscr{C}_{1-\delta}\left(v\sqrt{\sigma_{2% }}\right)\mathscr{C}_{1-\epsilon}\left(\mathrm{i}w\sqrt{\sigma_{1}+\sigma_{2}}% \right),

ⓘ

Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\mathrm{i}$ : imaginary unit, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\mathcal{K}(z;s,t)$ : kernel, $u$ , $v$ , $w$ and $\sigma_{1}$ , $\sigma_{2}$ : separation constants
Permalink:: http://dlmf.nist.gov/31.10.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

►where

\sigma_{1}

and

\sigma_{2}

are separation constants. … ►and

\sigma_{1}

and

\sigma_{2}

are separation constants. …

3: 29.18 Mathematical Applications

… ►

29.18.7

\frac{{\mathrm{d}}^{2}u_{3}}{{\mathrm{d}\gamma}^{2}}+(h-\nu(\nu+1)k^{2}{% \operatorname{sn}}^{2}\left(\gamma,k\right))u_{3}=0,

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter, $\gamma$ : sphero-conal coordinate and $u_{j}$ : solutions
Referenced by:: §29.18(i)
Permalink:: http://dlmf.nist.gov/29.18.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §29.18(i), §29.18 and Ch.29

►with separation constants

h

and

\nu

. …

4: 30.14 Wave Equation in Oblate Spheroidal Coordinates

… ►and

w_{2}

w_{3}

satisfy (30.13.10) and (30.13.11), respectively, with

\gamma^{2}=-\kappa^{2}c^{2}\leq 0

and separation constants

\lambda

and

\mu^{2}

. …

5: Bibliography S

… ►

J. A. Stratton, P. M. Morse, L. J. Chu, and R. A. Hutner (1941) Elliptic Cylinder and Spheroidal Wave Functions, Including Tables of Separation Constants and Coefficients. John Wiley and Sons, Inc., New York.

ⓘ

External Links:: MathReview (H. Bateman), ZentralBlatt
Referenced by:: §28.1, 7th item, Notations S, Notations S.
Encodings:: BibTeX

►

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and F. J. Corbató (1956) Spheroidal Wave Functions: Including Tables of Separation Constants and Coefficients. Technology Press of M. I. T. and John Wiley & Sons, Inc., New York.

ⓘ

External Links:: ISBN 0262190028, 0262692910, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: 1st item, Ch.30.
Encodings:: BibTeX

…

6: 30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►with

\gamma^{2}=\kappa^{2}c^{2}\geq 0

and separation constants

\lambda

and

\mu^{2}

. …

7: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►With

e

denoting here the elementary charge, the Coulomb potential between two point particles with charges

Z_{1}e,Z_{2}e

and masses

m_{1},m_{2}

separated by a distance

s

V(s)=Z_{1}Z_{2}e^{2}/(4\pi\varepsilon_{0}s)=Z_{1}Z_{2}\alpha\hbar c/s

, where

Z_{j}

are atomic numbers,

\varepsilon_{0}

is the electric constant,

\alpha

is the fine structure constant, and

\hbar

is the reduced Planck’s constant. The reduced mass is

m=m_{1}m_{2}/(m_{1}+m_{2})

, and at energy of relative motion

E

with relative orbital angular momentum

\ell\hbar

, the Schrödinger equation for the radial wave function

w(s)

is given by ►

33.22.1

\left(-\frac{\hbar^{2}}{2m}\left(\frac{{\mathrm{d}}^{2}}{{\mathrm{d}s}^{2}}-% \frac{\ell(\ell+1)}{s^{2}}\right)+\frac{Z_{1}Z_{2}\alpha\hbar c}{s}\right)w=Ew,

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\alpha$ : fine structure constant, $\ell$ : nonnegative integer, $Z_{k}$ : charges, $m_{k}$ : masses, $s$ : separation and $E$ : energy
Referenced by:: §33.22(i)
Permalink:: http://dlmf.nist.gov/33.22.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §33.22(i), §33.22 and Ch.33

…

8: 12.5 Integral Representations

… ►where the contour separates the poles of

\Gamma\left(t\right)

from those of

\Gamma\left(\tfrac{1}{2}+a-2t\right)

. … ►where the contour separates the poles of

\Gamma\left(t\right)

from those of

\Gamma\left(\tfrac{1}{2}-a-2t\right)

. …

9: 15.6 Integral Representations

… ►In (15.6.6) the integration contour separates the poles of

\Gamma\left(a+t\right)

and

\Gamma\left(b+t\right)

from those of

\Gamma\left(-t\right)

, and

(-z)^{t}

has its principal value. ►In (15.6.7) the integration contour separates the poles of

\Gamma\left(a+t\right)

and

\Gamma\left(b+t\right)

from those of

\Gamma\left(c-a-b-t\right)

and

\Gamma\left(-t\right)

, and

(1-z)^{t}

has its principal value. …

10: 13.4 Integral Representations

… ►where the contour of integration separates the poles of

\Gamma\left(a+t\right)

from those of

\Gamma\left(-t\right)

. … ►where the contour of integration separates the poles of

\Gamma\left(a+t\right)\Gamma\left(1+a-b+t\right)

from those of

\Gamma\left(-t\right)

. …