Asymptotic inversion of incomplete gamma functions

The normalized incomplete gamma functions $P(a,x)$ and $Q(a,x)$ are inverted for large values of the parameter a. That is, x-solutions of the equations

$$P(a,x) = p,\quad Q(a,x) = q,\quad p \in [0,1],q = 1 - p,$$

are considered, especially for large values of a. The approximations are obtained by using uniform asymptotic expansions of the incomplete gamma functions in which an error function is the dominant term. The inversion problem is started by inverting this error function term. Numerical results indicate that for obtaining an accuracy of four correct digits, the method can already be used for $a = 2$, although a is a large parameter. It is indicated that the method can be applied to other cumulative distribution functions.