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 the implied constant depends on $\epsilon$.

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


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

Taking the derivative of this expression we have:


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

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


 * $\displaystyle - \frac c{|z^{2d-1}|} < |\mathcal O(z^{1-2d})| < \frac c{|z^{2d-1}|},\quad |z| > 1$

it follows directly that the third term in $(1)$ is $\mathcal O(z^{-1})$, and we are done.