Approximation to Reciprocal times Derivative of Gamma Function

Theorem
Let $\Gamma$ denote the gamma function.

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


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

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

Proof
We have Stirling's Formula for the Gamma Function:


 * $\displaystyle \log \Gamma(z) = \left(z - \frac12\right)\log z - z + \frac{\log 2\pi}2 + \mathcal O \left({z^{-1}}\right)$

Taking the derivative of this expression we have:


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

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


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

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