W. Gautschi, Frank E. Harrris, and N. M. Temme

Department of Computer Sciences, Purdue University

Department of Physics, University of Utah

Quantum Theory Project, University of Florida

Center for Mathematics and Computer Science, Amsterdam

(Submitted to Appl. Math. Letters, January 2002).

An apparently new expansion of the exponential integral E_{1}in incomplete gamma functions is presented and shown to be a limiting case of a more general expansion given by Tricomi in 1950 without proof. This latter expansion is proved here by interpreting it as a "multiplication theorem". A companion result, not mentioned by Tricomi, holds for the complementary incomplete gamma function and can be applied to yield an expansion connecting E_{1}of different arguments. A general method is described for converting a power series into an expansion in incomplete gamma functions. In a special case, this provides an alternative derivation of Tricomi's expansion. Numerical properties of the new expansion for E_{1}are discussed.