Analytic saddle point approximation for ionization energy loss distributions

We present a saddle point approximation which gives closed form distributions for ionization energy losses valid at arbitrary relativistic velocities, provided that ionizing collisions are still the dominant energy loss mechanism. These analytic distributions are intended for use in simulations with relatively low computational overhead. These approximations generally reproduce the Vavilov most probable energy loss and full width at half maximum to better than 1% and 10%, respectively, with identical results obtained in the large energy loss Gaussian limit.