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

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

§33.10(ii) Large Positive $\eta$

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

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

In particular, for $\ell=0$,

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

Also,

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

§33.10(iii) Large Negative $\eta$

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

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

In particular, for $\ell=0$,

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

Also,

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