Dynamics of electrons in a half-space with cylindrical electro-static field

Motion of electrons in a half space with cylindrical electro-static field, which is associated with electron emission devices, is investigated. The equation of motion is given as a non-linear ordinary differential equation of the second order. Statistical behavior of electrons exhibits a dilatational invariance property of the system. An approximate formula of absorbed points distribution of the electrons is given by the solution of a Fredholm integral equation of the second kind. Since its kernel function is [0, 1) characteristrics function, we give its finite sum expression using the properties of B-spline, which is the iterated kernel function of its Neumann series solution.