Expansions of the Exponential Integral in Incomplete Gamma Functions

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 E1 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 E1 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 E1 are discussed.