# regular

(0.001 seconds)

## 1—10 of 51 matching pages

##### 1: 33.8 Continued Fractions
If we denote $u=\ifrac{F_{\ell}'}{F_{\ell}}$ and $p+\mathrm{i}q=\ifrac{{H^{+}_{\ell}}'}{{H^{+}_{\ell}}}$, then
$F_{\ell}=\pm(q^{-1}(u-p)^{2}+q)^{-1/2},$
$F_{\ell}'=uF_{\ell},$
$G_{\ell}=q^{-1}(u-p)F_{\ell},$
##### 2: 33.3 Graphics
###### §33.3(i) Line Graphs of the Coulomb Radial Functions $F_{\ell}\left(\eta,\rho\right)$ and $G_{\ell}\left(\eta,\rho\right)$ Figure 33.3.1: F ℓ ⁡ ( η , ρ ) , G ℓ ⁡ ( η , ρ ) with ℓ = 0 , η = - 2 . Magnify Figure 33.3.2: F ℓ ⁡ ( η , ρ ) , G ℓ ⁡ ( η , ρ ) with ℓ = 0 , η = 0 . Magnify Figure 33.3.3: F ℓ ⁡ ( η , ρ ) , G ℓ ⁡ ( η , ρ ) with ℓ = 0 , η = 2 . … Magnify
##### 3: 25.17 Physical Applications
Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect). It has been found possible to perform such regularizations by equating the divergent sums to zeta functions and associated functions (Elizalde (1995)).
##### 4: 33.23 Methods of Computation
The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ and $r$, respectively, and may be used to compute the regular and irregular solutions. … Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. … This implies decreasing $\ell$ for the regular solutions and increasing $\ell$ for the irregular solutions of §§33.2(iii) and 33.14(iii). … §33.8 supplies continued fractions for $F_{\ell}'/F_{\ell}$ and ${H^{\pm}_{\ell}}'/{H^{\pm}_{\ell}}$. Combined with the Wronskians (33.2.12), the values of $F_{\ell}$, $G_{\ell}$, and their derivatives can be extracted. …
##### 5: 33.2 Definitions and Basic Properties
###### §33.2(i) Coulomb Wave Equation
This differential equation has a regular singularity at $\rho=0$ with indices $\ell+1$ and $-\ell$, and an irregular singularity of rank 1 at $\rho=\infty$ (§§2.7(i), 2.7(ii)). …
###### §33.2(ii) Regular Solution $F_{\ell}\left(\eta,\rho\right)$
The function $F_{\ell}\left(\eta,\rho\right)$ is recessive (§2.7(iii)) at $\rho=0$, and is defined by … $F_{\ell}\left(\eta,\rho\right)$ is a real and analytic function of $\rho$ on the open interval $0<\rho<\infty$, and also an analytic function of $\eta$ when $-\infty<\eta<\infty$. …
##### 6: 33.5 Limiting Forms for Small $\rho$, Small $|\eta|$, or Large $\ell$
$F_{\ell}\left(\eta,\rho\right)\sim C_{\ell}\left(\eta\right)\rho^{\ell+1},$
$F_{\ell}'\left(\eta,\rho\right)\sim(\ell+1)C_{\ell}\left(\eta\right)\rho^{\ell}.$
$F_{\ell}\left(0,\rho\right)=\rho\mathsf{j}_{\ell}\left(\rho\right),$
$F_{0}\left(0,\rho\right)=\sin\rho,$
$F_{\ell}\left(\eta,\rho\right)\sim C_{\ell}\left(\eta\right)\rho^{\ell+1},$
##### 7: 33.24 Tables
• Abramowitz and Stegun (1964, Chapter 14) tabulates $F_{0}\left(\eta,\rho\right)$, $G_{0}\left(\eta,\rho\right)$, $F_{0}'\left(\eta,\rho\right)$, and $G_{0}'\left(\eta,\rho\right)$ for $\eta=0.5(.5)20$ and $\rho=1(1)20$, 5S; $C_{0}\left(\eta\right)$ for $\eta=0(.05)3$, 6S.

• ##### 8: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$
$F_{\ell}\left(\eta,\rho\right)=\sin\left({\theta_{\ell}}\left(\eta,\rho\right)% \right)+o\left(1\right),$
$F_{\ell}\left(\eta,\rho\right)\sim\dfrac{(2\ell+1)!C_{\ell}\left(\eta\right)}{% (2\eta)^{\ell+1}}(2\eta\rho)^{\ifrac{1}{2}}I_{2\ell+1}\left((8\eta\rho)^{% \ifrac{1}{2}}\right),$
$F_{0}\left(\eta,\rho\right)\sim e^{-\pi\eta}(\pi\rho)^{\ifrac{1}{2}}I_{1}\left% ((8\eta\rho)^{\ifrac{1}{2}}\right),$
$F_{0}'\left(\eta,\rho\right)\sim e^{-\pi\eta}(2\pi\eta)^{\ifrac{1}{2}}I_{0}% \left((8\eta\rho)^{\ifrac{1}{2}}\right),$
$F_{0}\left(\eta,\rho\right)=(\pi\rho)^{\ifrac{1}{2}}J_{1}\left((-8\eta\rho)^{% \ifrac{1}{2}}\right)+o\left({\left|\eta\right|}^{-\ifrac{1}{4}}\right),$
##### 9: 33.16 Connection Formulas
###### §33.16(i) $F_{\ell}$ and $G_{\ell}$ in Terms of $f$ and $h$
33.16.1 $F_{\ell}\left(\eta,\rho\right)=\dfrac{(2\ell+1)!C_{\ell}\left(\eta\right)}{(-2% \eta)^{\ell+1}}f\left(1/\eta^{2},\ell;-\eta\rho\right),$
###### §33.16(ii) $f$ and $h$ in Terms of $F_{\ell}$ and $G_{\ell}$ when $\epsilon>0$
33.16.4 $f\left(\epsilon,\ell;r\right)=\left(\frac{2}{\pi\tau}\frac{1-e^{-2\pi/\tau}}{A% (\epsilon,\ell)}\right)^{\ifrac{1}{2}}F_{\ell}\left(-1/\tau,\tau r\right),$
##### 10: 33.9 Expansions in Series of Bessel Functions
33.9.1 $F_{\ell}\left(\eta,\rho\right)=\rho\sum_{k=0}^{\infty}a_{k}\mathsf{j}_{\ell+k}% \left(\rho\right),$
33.9.3 $F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\frac{(2\ell+1)!}{(2% \eta)^{2\ell+1}}\rho^{-\ell}\*\sum_{k=2\ell+1}^{\infty}b_{k}t^{k/2}I_{k}\left(% \textstyle 2\sqrt{t}\right),$ $\eta>0$,
33.9.4 $F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\frac{(2\ell+1)!}{(2% \left|\eta\right|)^{2\ell+1}}\rho^{-\ell}\*\sum_{k=2\ell+1}^{\infty}\!\!b_{k}t% ^{k/2}J_{k}\left(\textstyle 2\sqrt{t}\right),$ $\eta<0$.