The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.

8> z 1 ds = 2 1 z Z 1 0 t 2z 1 e t 2 =2 dt: Artin [2] showed that (z) is the only smooth log-convex extension of the factorial function, so these integrals must coincide with the functions dened by Euler's limit (z) = lim n!1 1 2 3 n z(z + 1)(z + 2) (z + n) n z (2) The author acknowledges gratefully the support of US NSF grant DMS-9626829. 1 DRAFT and by Weierstrass' innite product 1=(z) = ze z 1 Y n=1 1 + z n<F18

This function evaluates the incomplete gamma functions in the normalised form P (a; x) = 1\Gamma (a) R x 0 t a\Gamma 1e\Gamma t dt Q(a; x) = 1\Gamma (a) R 1 x t a\Gamma 1e\Gamma t dt;

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Gamma Distribution Function subprogram GAMDIS calculates the gamma distribution function (incomplete gamma function) P (x; a) = 1\Gamma (a) Z x 0 e \Gamma t ta\Gamma 1 dt for real arguments x * 0 and a? 0. Structure: FUNCTION subprogramUser Entry Name:

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nag incomplete gamma (s14bac) nag incomplete gamma (s14bac) computes values for the incomplete gamma functions P (a, x) and