# §33.11 Asymptotic Expansions for Large $\rho$

For large $\rho$, with $\ell$ and $\eta$ fixed,

 33.11.1 $\mathop{{H^{\pm}_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=e^{\pm\mathrm{i}% \mathop{{\theta_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)}\sum_{k=0}^{\infty% }\frac{{\left(a\right)_{k}}{\left(b\right)_{k}}}{k!(\mp 2\mathrm{i}\rho)^{k}},$

where $\mathop{{\theta_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ is defined by (33.2.9), and $a$ and $b$ are defined by (33.8.3).

With arguments $(\eta,\rho)$ suppressed, an equivalent formulation is given by

 33.11.2 $\displaystyle\mathop{F_{\ell}\/}\nolimits$ $\displaystyle=g\mathop{\cos\/}\nolimits\mathop{{\theta_{\ell}}\/}\nolimits+f% \mathop{\sin\/}\nolimits\mathop{{\theta_{\ell}}\/}\nolimits,$ $\displaystyle\mathop{G_{\ell}\/}\nolimits$ $\displaystyle=f\mathop{\cos\/}\nolimits\mathop{{\theta_{\ell}}\/}\nolimits-g% \mathop{\sin\/}\nolimits\mathop{{\theta_{\ell}}\/}\nolimits,$
 33.11.3 $\displaystyle\mathop{F_{\ell}\/}\nolimits'$ $\displaystyle=\widehat{g}\mathop{\cos\/}\nolimits\mathop{{\theta_{\ell}}\/}% \nolimits+\widehat{f}\mathop{\sin\/}\nolimits\mathop{{\theta_{\ell}}\/}\nolimits,$ $\displaystyle\mathop{G_{\ell}\/}\nolimits'$ $\displaystyle=\widehat{f}\mathop{\cos\/}\nolimits\mathop{{\theta_{\ell}}\/}% \nolimits-\widehat{g}\mathop{\sin\/}\nolimits\mathop{{\theta_{\ell}}\/}\nolimits,$
 33.11.4 $\mathop{{H^{\pm}_{\ell}}\/}\nolimits=e^{\pm\mathrm{i}\mathop{{\theta_{\ell}}\/% }\nolimits}(f\pm\mathrm{i}g),$

where

 33.11.5 $\displaystyle f$ $\displaystyle\sim\sum_{k=0}^{\infty}f_{k},$ $\displaystyle g$ $\displaystyle\sim\sum_{k=0}^{\infty}g_{k},$ Defines: $f$: real part (locally) and $g$: imaginary part (locally) Symbols: $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $f_{k}$: coefficient and $g_{k}$: coefficient Permalink: http://dlmf.nist.gov/33.11.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 33.11
 33.11.6 $\displaystyle\widehat{f}$ $\displaystyle\sim\sum_{k=0}^{\infty}\widehat{f}_{k},$ $\displaystyle\widehat{g}$ $\displaystyle\sim\sum_{k=0}^{\infty}\widehat{g}_{k},$ Symbols: $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $f$: real part and $g$: imaginary part Permalink: http://dlmf.nist.gov/33.11.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 33.11
 33.11.7 $g\widehat{f}-f\widehat{g}=1.$ Symbols: $f$: real part and $g$: imaginary part A&S Ref: 14.5.8 Permalink: http://dlmf.nist.gov/33.11.E7 Encodings: TeX, pMML, png See also: Annotations for 33.11

Here $f_{0}=1$, $g_{0}=0$, $\widehat{f}_{0}=0$, $\widehat{g}_{0}=1-(\eta/\rho)$, and for $k=0,1,2,\dots$,

 33.11.8 $\displaystyle f_{k+1}$ $\displaystyle=\lambda_{k}f_{k}-\mu_{k}g_{k},$ $\displaystyle g_{k+1}$ $\displaystyle=\lambda_{k}g_{k}+\mu_{k}f_{k},$ $\displaystyle\widehat{f}_{k+1}$ $\displaystyle=\lambda_{k}\widehat{f}_{k}-\mu_{k}\widehat{g}_{k}-(f_{k+1}/\rho),$ $\displaystyle\widehat{g}_{k+1}$ $\displaystyle=\lambda_{k}\widehat{g}_{k}+\mu_{k}\widehat{f}_{k}-(g_{k+1}/\rho),$ Defines: $f_{k}$: coefficient (locally) and $g_{k}$: coefficient (locally) Symbols: $k$: nonnegative integer, $\rho$: nonnegative real variable, $f$: real part, $g$: imaginary part, $\lambda_{k}$: coefficient and $\mu_{k}$: coefficient Permalink: http://dlmf.nist.gov/33.11.E8 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 33.11

where

 33.11.9 $\displaystyle\lambda_{k}$ $\displaystyle=\frac{(2k+1)\eta}{(2k+2)\rho},$ $\displaystyle\mu_{k}$ $\displaystyle=\frac{\ell(\ell+1)-k(k+1)+\eta^{2}}{(2k+2)\rho}.$ Defines: $\lambda_{k}$: coefficient (locally) and $\mu_{k}$: coefficient (locally) Symbols: $k$: nonnegative integer, $\ell$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter Permalink: http://dlmf.nist.gov/33.11.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 33.11