Application of Coulomb wave functions to an orthogonal series associated with steady axisymmetric Euler flows

The regular Coulomb wave function of order zero is applied to an orthogonal series which is associated with the steady axisymmetric Euler equations. The series expansion of a function of bounded variation is proved to converge to the mean value of the left- and the right-hand limits of the original function at each point. In this proof, some complex functions related to the regular and the irregular Coulomb wave functions of orders zero and one are used.