3. Calcul analytique de \(S(R)\)
\[ S(R)=\frac1\pi\int e^{-(r_A+r_B)}\,d^3r. \]3.1 Coordonnées elliptiques prolates
\[ \mu=\frac{r_A+r_B}{R},\qquad \nu=\frac{r_A-r_B}{R}, \] \[ r_A=\frac R2(\mu+\nu),\qquad r_B=\frac R2(\mu-\nu), \] \[ d^3r=\frac{R^3}{8}(\mu^2-\nu^2)\,d\mu\,d\nu\,d\varphi. \]3.2 Séparation
\[ S=\frac1\pi\frac{R^3}{8}\int_0^{2\pi}d\varphi \int_1^\infty e^{-R\mu}d\mu \int_{-1}^{1}(\mu^2-\nu^2)d\nu. \] \[ \int_{-1}^{1}(\mu^2-\nu^2)d\nu=2\mu^2-\frac23. \] Donc \[ S=\frac{R^3}{2}\int_1^\infty e^{-R\mu}\left(\mu^2-\frac13\right)d\mu. \]3.3 Intégrales élémentaires
\[ \int_1^\infty e^{-R\mu}d\mu=\frac{e^{-R}}R, \] \[ \int_1^\infty \mu^2e^{-R\mu}d\mu =e^{-R}\left(\frac1R+\frac2{R^2}+\frac2{R^3}\right). \]
\[
\boxed{S(R)=e^{-R}\left(1+R+\frac{R^2}{3}\right)}.
\]