Analytic continuation of $\int_0^\infty t e^{-zt}\,dt$

Find an analytic continuation \(g ( z )\) of the function \(f ( z ) = \int _ { 0 } ^ { \infty } t e ^ { - z t } d t . \ f ( z )\) is defined only for \(z > 0\) . What is the domain of \(g ( z ) ?\)