Integrals for the LogGamma & polygamma fcns

[benorin]

I have proved that for $$\Re [z]>0,$$

$$\log \Gamma (z) = \int_{0}^{\infty} \left( z-1+\frac{1-e^{-(z-1)t}}{e^{-t}-1}\right) \frac{e^{-t}}{t}dt$$​

and I wish to justify differentiating under the integral sign (n+1)-times to give integral formulas for the polygamma functions. Let

$$f(t)=\left( z-1+\frac{1-e^{-(z-1)t}}{e^{-t}-1}\right) \frac{e^{-t}}{t}$$​

I have proved that if $$\Re [z]>0,\mbox{ and }|z|<\infty,$$ then

$$\lim_{t\rightarrow 0^+} f(t)=\frac{1}{2}z^2-\frac{3}{2}z+1,\mbox{ and }\lim_{t\rightarrow\infty} f(t)=0$$​

and if I recall correctly, it is sufficient that the integral in question be absolutely convergent to apply the differentiation rule.

EDIT: So how do I absolute convergence? Any hints?