Expansions of the exponential integral in incomplete gamma functions

✍ Scribed by W. Gautschi; F.E. Harris; N.M. Temme

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 289 KB
• Volume: 16
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/s0893-9659(03)90100-5

✦ Synopsis

apparently new expansion of the exponential integral El in incomplete gamma functions is presented and shown to be a limiting csse of a more general expansion given by Mcomi in 1950 without proof. This latter expansion is proved here by interpreting it as a "multiplication theorem". A companion result, not mentioned by 'l+icomi, holds for the complementary incomplete gamma function and can be applied to yield an expansion connecting El 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 El are discussed.