# Evaluating the log gamma integral ∫z0logΓ(x)dx\int_{0}^{z} \log \Gamma (x) \, \mathrm dx in terms of the Hurwitz zeta function

One way to evaluate $\displaystyle\int_{0}^{z} \log \Gamma(x) \, \mathrm dx$ is in terms of the Barnes G-function.

Another way is in terms of the Hurwitz zeta function.

I’ve been trying to prove the latter so that I can prove

My starting point is the generating function

Integrating both sides, I get

which implies

Then rearranging and integrating both sides from $0$ to $z$, I get

And then using the integral representation I get

Assuming I haven’t made any mistakes up to this point, how do I evaluate that limit?

To evaluate that limit, we can expand each function in a Laurent series at $s=0$ and use the following 3 facts about the Hurwitz zeta function:

$(1)$ http://dlmf.nist.gov/25.11 (25.11.3)
$(2)$ http://mathworld.wolfram.com/HurwitzZetaFunction.html (9)