# §33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$

## §33.10(i) Large $\rho$

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

 33.10.1 $\displaystyle\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle=\mathop{\sin\/}\nolimits\!\left(\mathop{{\theta_{\ell}}\/}% \nolimits\!\left(\eta,\rho\right)\right)+\mathop{o\/}\nolimits\!\left(1\right),$ $\displaystyle\mathop{G_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle=\mathop{\cos\/}\nolimits\!\left(\mathop{{\theta_{\ell}}\/}% \nolimits\!\left(\eta,\rho\right)\right)+\mathop{o\/}\nolimits\!\left(1\right),$
 33.10.2 $\mathop{{H^{\pm}_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)\sim\mathop{\exp\/% }\nolimits\!\left(\pm\mathrm{i}\mathop{{\theta_{\ell}}\/}\nolimits\!\left(\eta% ,\rho\right)\right),$

where $\mathop{{\theta_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ is defined by (33.2.9).

## §33.10(ii) Large Positive $\eta$

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

 33.10.3 $\displaystyle\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle\sim\dfrac{(2\ell+1)!\mathop{C_{\ell}\/}\nolimits\!\left(\eta% \right)}{(2\eta)^{\ell+1}}(2\eta\rho)^{\ifrac{1}{2}}\mathop{I_{2\ell+1}\/}% \nolimits\!\left((8\eta\rho)^{\ifrac{1}{2}}\right),$ $\displaystyle\mathop{G_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle\sim\dfrac{2(2\eta)^{\ell}}{(2\ell+1)!\mathop{C_{\ell}\/}% \nolimits\!\left(\eta\right)}(2\eta\rho)^{\ifrac{1}{2}}\mathop{K_{2\ell+1}\/}% \nolimits\!\left((8\eta\rho)^{\ifrac{1}{2}}\right).$

In particular, for $\ell=0$,

 33.10.4 $\displaystyle\mathop{F_{0}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle\sim e^{-\pi\eta}(\pi\rho)^{\ifrac{1}{2}}\mathop{I_{1}\/}% \nolimits\!\left((8\eta\rho)^{\ifrac{1}{2}}\right),$ $\displaystyle\mathop{G_{0}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle\sim 2e^{\pi\eta}\left(\ifrac{\rho}{\pi}\right)^{\ifrac{1}{2}}% \mathop{K_{1}\/}\nolimits\!\left((8\eta\rho)^{\ifrac{1}{2}}\right),$
 33.10.5 $\displaystyle\mathop{F_{0}\/}\nolimits'\!\left(\eta,\rho\right)$ $\displaystyle\sim e^{-\pi\eta}(2\pi\eta)^{\ifrac{1}{2}}\mathop{I_{0}\/}% \nolimits\!\left((8\eta\rho)^{\ifrac{1}{2}}\right),$ $\displaystyle\mathop{G_{0}\/}\nolimits'\!\left(\eta,\rho\right)$ $\displaystyle\sim-2e^{\pi\eta}\left(\ifrac{2\eta}{\pi}\right)^{\ifrac{1}{2}}% \mathop{K_{0}\/}\nolimits\!\left((8\eta\rho)^{\ifrac{1}{2}}\right).$

Also,

 33.10.6 $\displaystyle\mathop{{\sigma_{0}}\/}\nolimits\!\left(\eta\right)$ $\displaystyle=\eta(\mathop{\ln\/}\nolimits\eta-1)+\tfrac{1}{4}\pi+\mathop{o\/}% \nolimits\!\left(1\right),$ $\displaystyle\mathop{C_{0}\/}\nolimits\!\left(\eta\right)$ $\displaystyle\sim(2\pi\eta)^{1/2}e^{-\pi\eta}.$

## §33.10(iii) Large Negative $\eta$

As $\eta\to-\infty$ with $\rho$ fixed,

 33.10.7 $\displaystyle\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle=\dfrac{(2\ell+1)!\mathop{C_{\ell}\/}\nolimits\!\left(\eta\right)% }{(-2\eta)^{\ell+1}}\left((-2\eta\rho)^{\ifrac{1}{2}}\*\mathop{J_{2\ell+1}\/}% \nolimits\!\left((-8\eta\rho)^{\ifrac{1}{2}}\right)+\mathop{o\/}\nolimits\!% \left({\left|\eta\right|^{\ifrac{1}{4}}}\right)\right),$ $\displaystyle\mathop{G_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle=-\dfrac{\pi(-2\eta)^{\ell}}{(2\ell+1)!\mathop{C_{\ell}\/}% \nolimits\!\left(\eta\right)}\left((-2\eta\rho)^{\ifrac{1}{2}}\*\mathop{Y_{2% \ell+1}\/}\nolimits\!\left((-8\eta\rho)^{\ifrac{1}{2}}\right)+\mathop{o\/}% \nolimits\!\left({\left|\eta\right|^{\ifrac{1}{4}}}\right)\right).$

In particular, for $\ell=0$,

 33.10.8 $\displaystyle\mathop{F_{0}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle=(\pi\rho)^{\ifrac{1}{2}}\mathop{J_{1}\/}\nolimits\!\left((-8\eta% \rho)^{\ifrac{1}{2}}\right)+\mathop{o\/}\nolimits\!\left({\left|\eta\right|^{-% \ifrac{1}{4}}}\right),$ $\displaystyle\mathop{G_{0}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle=-(\pi\rho)^{\ifrac{1}{2}}\mathop{Y_{1}\/}\nolimits\!\left((-8% \eta\rho)^{\ifrac{1}{2}}\right)+\mathop{o\/}\nolimits\!\left({\left|\eta\right% |^{-\ifrac{1}{4}}}\right).$
 33.10.9 $\displaystyle\mathop{F_{0}\/}\nolimits'\!\left(\eta,\rho\right)$ $\displaystyle=(-2\pi\eta)^{\ifrac{1}{2}}\mathop{J_{0}\/}\nolimits\!\left((-8% \eta\rho)^{\ifrac{1}{2}}\right)+\mathop{o\/}\nolimits\!\left({\left|\eta\right% |^{\ifrac{1}{4}}}\right),$ $\displaystyle\mathop{G_{0}\/}\nolimits'\!\left(\eta,\rho\right)$ $\displaystyle=-(-2\pi\eta)^{\ifrac{1}{2}}\mathop{Y_{0}\/}\nolimits\!\left((-8% \eta\rho)^{\ifrac{1}{2}}\right)+\mathop{o\/}\nolimits\!\left({\left|\eta\right% |^{\ifrac{1}{4}}}\right).$

Also,

 33.10.10 $\displaystyle\mathop{{\sigma_{0}}\/}\nolimits\!\left(\eta\right)$ $\displaystyle=\eta(\mathop{\ln\/}\nolimits\!\left(-\eta\right)-1)-\tfrac{1}{4}% \pi+\mathop{o\/}\nolimits\!\left(1\right),$ $\displaystyle\mathop{C_{0}\/}\nolimits\!\left(\eta\right)$ $\displaystyle\sim(-2\pi\eta)^{1/2}.$