CoulombF is the regular Coulomb wave function. It satisfies the differential equation

$\mathrm{CoulombF}\left(0\,0\,x\right)$

$\sin \left(x\right)$

$\mathrm{CoulombF}\left(3\,0\,x\right)$

$\frac{\sqrt{\mathrm{\pi }x}\left(\cos \left(x\right)x^3-6\sin \left(x\right)x^2-15\cos \left(x\right)x+15\sin \left(x\right)\right)}{\sqrt{\mathrm{\pi }}{x}^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

$\mathrm{CoulombF}\left(2.4\,1.7\,-3.2+\mathrm{I}\right)$

$0.001110433908$

$\mathrm{diff}\left(\mathrm{CoulombF}\left(L\,n\,x\right)\,x\right)$

$\frac{\left(\frac{{\left(L+1\right)}^2}{x}+n\right)\mathrm{CoulombF}\left(L\,n\,x\right)}{L+1}-\frac{\sqrt{{\left(L+1\right)}^2+n^2}\mathrm{CoulombF}\left(L+1\,n\,x\right)}{L+1}$

$\mathrm{Coulomb\_ODE}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\left(1-\frac{2n}{x}-\frac{L\left(L+1\right)}{x^2}\right)y\left(x\right)$

$\mathrm{Coulomb\_ODE}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+\left(1-\frac{2n}{x}-\frac{L\left(L+1\right)}{x^2}\right)y\left(x\right)$

$\mathrm{odetest}\left(y\left(x\right)=\mathrm{CoulombF}\left(L\,n\,x\right)\,\mathrm{Coulomb\_ODE}\right)$

$0$

$\mathrm{dsolve}\left(\mathrm{Coulomb\_ODE}\right)$

$y\left(x\right)=\mathrm{\_C1}\mathrm{WhittakerM}\left(\mathrm{I}n\,L+\frac{1}{2}\,2\mathrm{I}x\right)+\mathrm{\_C2}\mathrm{WhittakerW}\left(\mathrm{I}n\,L+\frac{1}{2}\,2\mathrm{I}x\right)$

$\mathrm{CoulombF}\left(L\,n\,x\right)=\mathrm{convert}\left(\mathrm{CoulombF}\left(L\,n\,x\right)\,\mathrm{WhittakerM}\right)$

$\mathrm{CoulombF}\left(L\,n\,x\right)=\frac{2^L\left|\mathrm{\Gamma }\left(1+L+\mathrm{I}n\right)\right|x^{L+1}\mathrm{WhittakerM}\left(\mathrm{I}n\,L+\frac{1}{2}\,2\mathrm{I}x\right)}{{\ⅇ}^{\frac{n\mathrm{\pi }}{2}}\mathrm{\Gamma }\left(2+2L\right){\left(2\mathrm{I}x\right)}^{L+1}}$

Abramowitz, M., and Stegun, I. eds. Handbook of Mathematical Functions. New York: Dover publications.