库伦函数

库伦函数(Couloumb function)是下列常微分方程的解

Coulomb F function plot

Coulomb G function plot

${\displaystyle {\frac {d^{2}w(\rho )}{d\rho ^{2}}}+(1-{\frac {2\eta }{\rho }}-{\frac {l*(l+1)}{\rho ^{2}}})*w(\rho )=0}$

此方程的两个解为：

${\displaystyle F_{\ell }(\eta ,\rho )}$ 和 ${\displaystyle G_{\ell }(\eta ,\rho )}$, 以 合流超几何函数表示^[1][2]

${\displaystyle F_{\ell }(\eta ,\rho )={\frac {2^{\ell }e^{-\pi \eta /2}|\Gamma (\ell +1+i\eta )|}{(2\ell +1)!}}\rho ^{\ell +1}e^{\mp i\rho }M(\ell +1\mp i\eta ,2\ell +2,\pm 2i\rho )\,.}$