Digamma function

Remarks

Example: from $\Gamma(z+1) = z\,\Gamma(z)$, taking logarithmic derivatives gives $\psi(z+1) = \psi(z) + \frac1z$. With $\psi(1) = \Gamma'(1) = -\gamma$, where $\gamma$ is the Euler–Mascheroni constant, this gives $\psi(n+1) = H_n - \gamma$ for every integer $n \ge 1$, where $H_n = 1 + \frac12 + \cdots + \frac1n$ is the harmonic number. Another value is $\psi(\frac12) = -\gamma - 2\ln 2$.

Non-example of a valid argument: $z = 0$ is a pole of $\Gamma$, so $\psi$ is not defined there. In fact $\psi$ has a simple pole with residue $-1$ at each of $0, -1, -2, \ldots$.

The function $\psi$ has no other poles, because $\Gamma$ has no zeros.