The irregular radial Coulomb wave function is expanded in a convergent series of Bessel functions in which the coefficients are expressed in powers of the energy and the argument of the Bessel functions depends on the radius only and not on the energy. This formulation is suitable for low-energy particles. The corresponding expansion for the regular Coulomb function can be deduced from previous work by Tricomi.

In a previous paper (3) one of us reported an expansion for the exponential integral

in terms of Bessel functions. In this note, we shall obtain the more general formula

where n is any positive integer, γ is Euler's constant and

It reduces to that in (3) when n = 1.

The irregular radial Coulomb wave function in repulsive fields is expressed, for any angular momentum value, in series of Bessel functions of arbitrary order. Previous expansions, obtained for angular momenta zero and one, can now be obtained as special cases. The expansion can be most conveniently used for high-energy particles.

An expansion of the Whittaker function Mk, m(z) in terms of functions of the same kind is obtained. It generalizes previous results and is used to derive expansions of the second Whittaker function Wk,m (z) in terms of the M functions for the cases m = 0 and ½. The case m = 0 provides expansions for the Poiseuille and the exponential integral functions. The case m = ½ provides an expansion for the irregular Coulomb wave function having angular momentum zero in terms of the regular Coulomb functions.

An expansion of the Whittaker function Wk, m(z) in terms of Bessel functions is given for 2m an integer or zero. Simple formulae are derived for the cases of m = 0, ½, 1 and . The case m = 0 provides expansions for the exponential, sine, cosine and similar integrals. The cases m = ½ and provide expansions for the irregular Coulomb wave functions having angular momenta zero and one.