# §33.5 Limiting Forms for Small $\rho$, Small $|\eta|$, or Large $\ell$

## §33.5(i) Small $\rho$

As $\rho\to 0$ with $\eta$ fixed,

 33.5.1 $\displaystyle\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle\sim\mathop{C_{\ell}\/}\nolimits\!\left(\eta\right)\rho^{\ell+1},$ $\displaystyle\mathop{F_{\ell}\/}\nolimits'\!\left(\eta,\rho\right)$ $\displaystyle\sim(\ell+1)\mathop{C_{\ell}\/}\nolimits\!\left(\eta\right)\rho^{% \ell}.$
 33.5.2 $\displaystyle\mathop{G_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle\sim\frac{\rho^{-\ell}}{(2\ell+1)\mathop{C_{\ell}\/}\nolimits\!% \left(\eta\right)},$ $\ell=0,1,2,\dots$, $\displaystyle\mathop{G_{\ell}\/}\nolimits'\!\left(\eta,\rho\right)$ $\displaystyle\sim-\frac{\ell\rho^{-\ell-1}}{(2\ell+1)\mathop{C_{\ell}\/}% \nolimits\!\left(\eta\right)},$ $\ell=1,2,3,\dots$.

## §33.5(ii) $\eta=0$

 33.5.3 $\displaystyle\mathop{F_{\ell}\/}\nolimits\!\left(0,\rho\right)$ $\displaystyle=\rho\mathop{\mathsf{j}_{\ell}\/}\nolimits\!\left(\rho\right),$ $\displaystyle\mathop{G_{\ell}\/}\nolimits\!\left(0,\rho\right)$ $\displaystyle=-\rho\mathop{\mathsf{y}_{\ell}\/}\nolimits\!\left(\rho\right).$

Equivalently,

 33.5.4 $\displaystyle\mathop{F_{\ell}\/}\nolimits\!\left(0,\rho\right)$ $\displaystyle=(\pi\rho/2)^{1/2}\mathop{J_{\ell+\frac{1}{2}}\/}\nolimits\!\left% (\rho\right),$ $\displaystyle\mathop{G_{\ell}\/}\nolimits\!\left(0,\rho\right)$ $\displaystyle=-(\pi\rho/2)^{1/2}\mathop{Y_{\ell+\frac{1}{2}}\/}\nolimits\!% \left(\rho\right).$

For the functions $\mathop{\mathsf{j}\/}\nolimits$, $\mathop{\mathsf{y}\/}\nolimits$, $\mathop{J\/}\nolimits$, $\mathop{Y\/}\nolimits$ see §§10.47(ii), 10.2(ii).

 33.5.5 $\displaystyle\mathop{F_{0}\/}\nolimits\!\left(0,\rho\right)$ $\displaystyle=\mathop{\sin\/}\nolimits\rho,$ $\displaystyle\mathop{G_{0}\/}\nolimits\!\left(0,\rho\right)$ $\displaystyle=\mathop{\cos\/}\nolimits\rho,$ $\displaystyle\mathop{{H^{\pm}_{0}}\/}\nolimits\!\left(0,\rho\right)$ $\displaystyle=e^{\pm i\rho}.$
 33.5.6 $\mathop{C_{\ell}\/}\nolimits\!\left(0\right)=\frac{2^{\ell}\ell!}{(2\ell+1)!}=% \frac{1}{(2\ell+1)!!}.$

## §33.5(iii) Small $|\eta|$

 33.5.7 $\mathop{{\sigma_{0}}\/}\nolimits\!\left(\eta\right)\sim-\EulerConstant\eta,$ $\eta\to 0$,

where $\EulerConstant$ is Euler’s constant (§5.2(ii)).

## §33.5(iv) Large $\ell$

As $\ell\to\infty$ with $\eta$ and $\rho$ ($\neq 0$) fixed,

 33.5.8 $\displaystyle\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle\sim\mathop{C_{\ell}\/}\nolimits\!\left(\eta\right)\rho^{\ell+1},$ $\displaystyle\mathop{G_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle\sim\dfrac{\rho^{-\ell}}{(2\ell+1)\mathop{C_{\ell}\/}\nolimits\!% \left(\eta\right)},$
 33.5.9 $\mathop{C_{\ell}\/}\nolimits\!\left(\eta\right)\sim\dfrac{e^{-\pi\eta/2}}{(2% \ell+1)!!}\sim e^{-\pi\eta/2}\dfrac{e^{\ell}}{\sqrt{2}(2\ell)^{\ell+1}}.$