Prove that

**Solution**

We state two lemmata first.

**Lemma 1: **Let . It holds that

*Proof: *A standard proof can be found through Fourier Series. One can expand in Fourier series the function . Another way to prove the identity is to begin from the Weierstrass product , that is . Taking log on both sides we have that

Differentiating we have

**Lemma 2: **It holds that

*Proof: *Just differentiate the above identity.

Apply the above lemmata , we have for the initial integral that

Another related Fourier series is the following:

Proof:We are using the reflection formula of the digamma function to extract the result. We have successively:and the result follows.

Of course,