A Hankel Transform approach to Doppler broadening

The kernel broadening approach to Doppler broadening has been shown to be equivalent to the convolution of a function related to the unheated cross section and Gaussian distribution function. This convolution may be expressed as the inverse Fourier transform of the product of the Fourier transforms of the convolution arguments. By recognizing that the unheated cross section function is an odd function, the resulting equations are reduced to a Hankel Transform of order $\alpha =\frac{1}{2}$. A discretized version of the Hankel Transform is applied to produce a set of equations that are independent of the interpolation scheme used in the pointwise representation of the cross sections. The heated cross section at a given energy is consequently calculated as the sum of the contributions from each of these pointwise intervals and each contribution is represented as a separate Hankel Transform.