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

## Β§33.5(i) Small $\rho$

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

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

## Β§33.5(ii) $\eta=0$

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

Equivalently,

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

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

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

## Β§33.5(iii) Small $|\eta|$

 33.5.7 ${\sigma_{0}}\left(\eta\right)\sim-\gamma\eta,$ $\eta\to 0$, β

where $\gamma$ is Eulerβs constant (Β§5.2(ii)).

## Β§33.5(iv) Large $\ell$

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

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