 Note that when "m" = 0, the second path does not contain any pole, and so the integral must vanish identically,

if either "p" < "q", or "p" = "q" and |"z"| < 1.Similarly, if the integral converges when evaluated along the third path above, and if no confluent poles appear among the Γ(1 − "a""k" + "s"), "k" = 1, 2, ..., "n", then the G-function can be expressed as