Asymptotic and Series Expansion of the Generalized Exponential Integrals

摘要

The generalization of the exponential integral is reviewed and the more important formulas restated. The asymptotic expansion of the generalized exponential integral is then developed with some interesting properties noted. The more difficult and complicated derivation of the series expansion is given in detail. In the process of this derivation a novel connection with the gamma function and its derivatives emerges, leading to an elegant formula for the series expansion of the generalized exponential integral. Inherent in this development is a sequence of numbers dependent only on Riemann zeta function of even integer values of the argument. Just as for the Riemann zeta functions it is shown that the numerical values of these numbers lie between 1 and 2. (Author)