The optimized mathematical treatment on the correct cutoff for integrals with the Rutherford scattering cross section

The traditional cutoff variables for the Rutherford scattering cross section in plasma physics, such as scattering angle θ and impact parameter b, are incorrect cutoff variables for lacking the necessary cutoff on relative speed g. The correct cutoff variable has proven to be the magnitude of the velocity change of a test particle δ=|v′−v|, where v and v′ are velocities of the test particle before and after a collision, respectively. Neither θ=0 nor b=∞ is the singularity point for the Rutherford scattering cross section. The singularity point of the Rutherford scattering cross section is correctly described as δ=0. A complete optimized mathematical treatment for implementing the correct cutoff δ≥δmin is presented in this paper. The treatment leads to the finding of a set of functions qn(k)(0,u) whose properties are like the simple power functions un. The arbitrary higher order of the Fokker–Planck coefficients, transition moments, and Rosenbluth potentials are greatly simplified with this set of functions. The integral property of qn(k)(ymin,u) has greatly simplified the calculation of the energy transfer rates. The result of energy transfer rates leads to the derivation of the most important statistical variable CF, which represents the absolute measure for binary collisions. The absolute and universal correct cutoff is proven to be CF≥CFmin. All the cutoff constants are obtained exactly in the form of incomplete Gamma functions of different orders. In particular, the so-called Coulomb logarithm ln Λ should be replaced by the exact form—the zeroth order incomplete Gamma function 12Γ(0,CFminkBTeff).