Approximation to Reciprocal times Derivative of Gamma Function

Theorem
Let $\Gamma$ denote the gamma function.

For all $z \in \C$ with $|\arg(z)| < \pi - \epsilon, |z| > 1$:


 * $\dfrac {\Gamma' \left({z}\right)} {\Gamma \left({z}\right)} = \log z + \mathcal O \left({z^{-1}}\right)$

where:
 * $\mathcal O \left({z^{-1}}\right)$ denotes big-O notation
 * the implied constant depends on $\epsilon$.

Proof
We have Stirling's Formula for Gamma Function:


 * $\log \Gamma(z) = \left({z - \dfrac 1 2}\right) \log z - z + \dfrac {\log 2 \pi} 2 + \mathcal O \left({z^{-1}}\right)$

Taking the derivative of this expression we have:


 * $(1): \quad \dfrac{\Gamma' \left({z}\right)} {\Gamma \left({z}\right)} = \log z - \dfrac 1 {2z} + \dfrac {\mathrm d} {\mathrm dz} \mathcal O \left({z^{-1}}\right)$

Since there is $c(\epsilon) > 0$ such that:


 * $- \dfrac c{|z^{-1}|} < |\mathcal O \left({z^{-1}}\right)| < \dfrac c{|z^{-1}|},\quad |z| > 1$

it follows directly that the third term in $(1)$ is $\mathcal O \left({z^{-1}}\right)$.