Uniform Asymptotic Expansions of the Incomplete Gamma Functions and the Incomplete Beta Function

Abstract

New asymptotic expansions are derived for the incomplete gamma functions and the incomplete beta function.In each case the expansion contains the complementary error function and an asymptotic series.The expansions are uniformly valid with respect to certain domains of the parameters.1. Introduction.The incomplete gamma functions are defined byThe parameters may be complex; but here we suppose a and x to be real, where a > 0 and x > 0. Auxiliary functions areand from the definitions it follows that(1 -3) J(a, x) + T(a, x) = V(a), P(a, x) + Q(a, x) = 1.For large values of x we have the well-known asymptotic expansion, r(a, x) ~ xa"Vx{T + (a -l)/x + (a -l)(a -2)/x2 + ...}.See for instance Dingle [1] or Olver [3].If both x and a are large, this expansion is not useful, unless a = o(x).For large values of a, we can better use the function y(a, x).From (1.1) we obtain the elementary result oo y(a, x) = e-xxT(a) x"/r(a + n + 1).n = 0This series converges for every finite x.It is useful for a - and x = o(a), since under this condition the series has an asymptotic character.Expansions with a more uniform character are given by Tricomi [4], who found among others y(a + 1, a + y(2a)'A)lT(a + 1) = Vi erfc(-j) -|(2/W)y2(l + y2)exp(-y2)(1 4) + 0(a~l), y, a real, a -* + .This expansion is uniformly valid in y on compact intervals of R. The function erfc is the complementary error function defined by