In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,
where  is the partial scattering amplitude and  are the Legendre polynomials. The partial amplitude can be expressed via the partial wave S-matrix element  () and the scattering phase shift  as
Then the total cross section
can be expanded as
is the partial cross section. The total cross section is also equal to  due to optical theorem.
For , we can write