# eta function

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## 11—20 of 72 matching pages

##### 11: 20.3 Graphics Figure 20.3.2: θ 1 ⁡ ( π ⁢ x , q ) , 0 ≤ x ≤ 2 , q = 0. …Here q Dedekind = e - π ⁢ y 0 = 0.19 approximately, where y = y 0 corresponds to the maximum value of Dedekind’s eta function η ⁡ ( i ⁢ y ) as depicted in Figure 23.16.1. Magnify
##### 13: 33.12 Asymptotic Expansions for Large $\eta$
###### §33.12(i) Transition Region
Then as $\eta\to\infty$, …
##### 14: 33.4 Recurrence Relations and Derivatives
###### §33.4 Recurrence Relations and Derivatives
Then, with $X_{\ell}$ denoting any of $F_{\ell}\left(\eta,\rho\right)$, $G_{\ell}\left(\eta,\rho\right)$, or ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$, …
##### 15: 33.13 Complex Variable and Parameters
###### §33.13 Complex Variable and Parameters
The functions $F_{\ell}\left(\eta,\rho\right)$, $G_{\ell}\left(\eta,\rho\right)$, and ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$ may be extended to noninteger values of $\ell$ by generalizing $(2\ell+1)!=\Gamma\left(2\ell+2\right)$, and supplementing (33.6.5) by a formula derived from (33.2.8) with $U\left(a,b,z\right)$ expanded via (13.2.42). These functions may also be continued analytically to complex values of $\rho$, $\eta$, and $\ell$. …
33.13.1 $C_{\ell}\left(\eta\right)=2^{\ell}e^{\mathrm{i}{\sigma_{\ell}}\left(\eta\right% )-(\pi\eta/2)}\Gamma\left(\ell+1-\mathrm{i}\eta\right)/\Gamma\left(2\ell+2% \right),$
##### 17: 33.1 Special Notation
The main functions treated in this chapter are first the Coulomb radial functions $F_{\ell}\left(\eta,\rho\right)$, $G_{\ell}\left(\eta,\rho\right)$, ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$ (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions $f\left(\epsilon,\ell;r\right)$, $h\left(\epsilon,\ell;r\right)$, $s\left(\epsilon,\ell;r\right)$, $c\left(\epsilon,\ell;r\right)$ (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions. …
• Greene et al. (1979):

$f^{(0)}(\epsilon,\ell;r)=f\left(\epsilon,\ell;r\right)$, $f(\epsilon,\ell;r)=s\left(\epsilon,\ell;r\right)$, $g(\epsilon,\ell;r)=c\left(\epsilon,\ell;r\right)$.

• ##### 18: 33.5 Limiting Forms for Small $\rho$, Small $|\eta|$, or Large $\ell$
###### §33.5(ii) $\eta=0$
33.5.6 $C_{\ell}\left(0\right)=\frac{2^{\ell}\ell!}{(2\ell+1)!}=\frac{1}{(2\ell+1)!!}.$
##### 19: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$
###### §33.10(i) Large $\rho$
33.10.2 ${H^{\pm}_{\ell}}\left(\eta,\rho\right)\sim\exp\left(\pm\mathrm{i}{\theta_{\ell% }}\left(\eta,\rho\right)\right),$
##### 20: 33.3 Graphics
###### §33.3(ii) Surfaces of the Coulomb Radial Functions$F_{0}\left(\eta,\rho\right)$ and $G_{0}\left(\eta,\rho\right)$ Figure 33.3.8: G 0 ⁡ ( η , ρ ) , - 2 ≤ η ≤ 2 , 0 < ρ ≤ 5 . Magnify 3D Help