# Coulomb Functions¶

## Normalized Hydrogenic Bound States¶

gsl_sf_hydrogenicR_1(Z, r)

This routine computes the lowest-order normalized hydrogenic bound state radial wavefunction $$R_1 := 2 Z \sqrt{Z} \exp(-Z r)$$.

gsl_sf_hydrogenicR(n, l, Z, r)

This routine computes the $$n$$-th normalized hydrogenic bound state radial wavefunction,

$R_n := 2 (Z^{3/2}/n^2) \sqrt{(n-l-1)!/(n+l)!} \exp(-Z r/n) (2Zr/n)^l L^{2l+1}_{n-l-1}(2Zr/n).$

where $$L^a_b(x)$$ is the generalized Laguerre polynomial (see Laguerre Functions). The normalization is chosen such that the wavefunction $$\psi$$ is given by $$\psi(n,l,r) = R_n Y_{lm}$$.

## Coulomb Wave Function Normalization Constant¶

The Coulomb wave function normalization constant is defined in Abramowitz 14.1.7.

gsl_sf_coulomb_CL(L, eta)

This function computes the Coulomb wave function normalization constant $$C_L(\eta)$$ for $$L > -1$$.