Incomplete Gamma Function

The "complete" gamma function  can be generalized to the incomplete gamma function  such that . This "upper" incomplete gamma function is given by

(1)

For  an integer 

			(2)
			(3)

where  is the exponential sum function. It is implemented as Gamma[a, z] in the Wolfram Language.

The special case of  can be expressed in terms of the subfactorial  as

(4)

The incomplete gamma function  has continued fraction

(5)

(Wall 1948, p. 358).

The lower incomplete gamma function is given by

			(6)
			(7)
			(8)

where  is the confluent hypergeometric function of the first kind. For  an integer ,

			(9)
			(10)

It is implemented as Gamma[a, 0, z] in the Wolfram Language.

By definition, the lower and upper incomplete gamma functions satisfy

(11)

The exponential integral  is closely related to the incomplete gamma function  by

(12)

Therefore, for real ,

(13)

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 260, 1972.

Arfken, G. "The Incomplete Gamma Function and Related Functions." §10.5 in Mathematical Methods for Physicists, 3rd ed.Orlando, FL: Academic Press, pp. 565-572, 1985.

Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.