  The evaluation of the electron-ion bremsstrahlung cross section exact to all
orders in the Coulomb potential is computationally expensive due to the
appearance of hypergeometric functions. Therefore tabulations are widely used.
Here, we provide an approximate formula for the non-relativistic dipole process
valid for all applicable relative velocities and photon energies. Its validity
spans from the Born- to the classical regime and from soft-photon emission to
the kinematic endpoint. The error remains below 3% (and widely below 1%) except
at an isolated region of hard-photon emission at the quantum-to-classical
crossover. We use the formula to obtain the thermally averaged emission
spectrum and cooling function in a Maxwellian plasma and demonstrate that they
are accurate to better than 2%.
