I was messing around with the digamma function the other day, and I discovered this identity: $$\psi\left(\frac ab\right)=\sum_{\substack{\large\rho^b=1\\\large\rho\ne1}}(\rho^a-1)\ln(1-\bar\rho)-\gamma$$ when $0<\dfrac ab\le1$. It's unusual in that it sums over the $b$-eth roots of unity (which I don't see very often). (Note that $\bar\rho=\rho^{-1}$.) It also gives explicit values of the digamma function for all rational arguments, but no irrational ones.
Another thing that's interesting is that it's well-defined. For example, I know that $\psi\left(\dfrac12\right)=\psi\left(\dfrac24\right)$, but that's not immediately obvious if I plug those values into the RHS. It's also not immediately obvious—looking at the RHS—that the function is continuous.
Example: $\psi\left(\dfrac14\right)=(i-1)\ln(1+i)+(-1-1)\ln(1+1)+(-i-1)\ln(1-i)-\gamma=\\-3\ln2-\dfrac\pi2-\gamma$.
Since $H_n=\dfrac1n+\gamma+\psi(n)$, we can get a similar identity for the Harmonic numbers.
Is this identity well-known? Does it have a name?
(And, if so, is there a way to integrate it, so that I get a similar formula for the Gamma function?)