Mathieu functions and coulomb spheroidal functions in the electrostatic probe theory

A spherical probe placed in a slowly moving collisional plasma with a large Debye length $\lambda_{\mathrm D}\to\infty$ is considered. The partial differential equation describing the electron concentration around the probe is reduced to two ordinary differential equations, namely, to the equation for Coulomb spheroidal functions and Mathieu’s modified equation with the parameter $a$ of the latter related to the eigenvalue $\lambda$ of the former by the relation $a=\lambda+1/4$. It is shown that the solutions of Mathieu’s equation are Mathieu functions of half-integer order, which are expressed as series in terms of spherical Bessel functions and series of products of Bessel functions. These Mathieu functions are numerically constructed for Mathieu’s modified and usual equations.