Uniform bounds for the complementary incomplete Gamma function

We prove upper and lower bounds for the complementary incomplete gamma function (a, z) with complex parameters a and z . Our bounds are refined within the circular hyperboloid of one sheet {(a, z) : |z| > c|a -1|} with a real and z complex. Our results show that within the hyperboloid, |(a, z)| is of order |z| a-1 e -Re(z) , and extends an upper estimate of Natalini and Palumbo to complex values of z .