The problem is computing the integral expression $$f(t)=\int_0^{\infty}\lambda^{t-1}e^{-\lambda z}d\lambda$$ where z is a complex variable with $Re(z)\ge 0$. Is it correct to substitute $w=\lambda z$ in the integral and write down $f(t)$ as $$f(t)=z^{-t}\int_0^{\infty}w^{t-1}e^{-w}dw$$ where w is a complex variable?
Note that while in the first integral $\lambda$ is a real variable which approaches positive infinity of real numbers, in the second integral $w$ is a complex variable and the integral is only meaningful if its value is independent from how $w$ approaches the infinity point of the extended complex plane.