An improved asymptotic series for an integral representing transient current on an infinite cylindrical antenna caused by a step voltage source, applied at the gap, is given. A generating function for obtaining the general coefficient is derived. The method is applicable to types of integrals whose integrands consist of a product of a slowly varying function and a rapidly varying function. A simple accurate formula obtained before, by the averaging method, is shown to be the leading term of this infinite series. The averaging used in evaluating integrals is shown to improve the accuracy of the leading term.