Approximation to Reciprocal times Derivative of Gamma Function

Theorem
Let $\Gamma$ denote the gamma function.

For all $z \in \C$ such that $\cmod {\map \arg z} < \pi - \epsilon, \cmod z > 1$:


 * $\dfrac {\map {\Gamma'} z} {\map \Gamma z} = \ln z + \map {\OO_\epsilon} {z^{-1} }$

where:
 * $\map \OO {z^{-1} }$ denotes big-O notation
 * the implied constant depends on $\epsilon$.

Proof
From Stirling's Formula for Gamma Function:


 * $\ln \map \Gamma z = \paren {z - \dfrac 1 2} \ln z - z + \dfrac {\ln 2 \pi} 2 + \map \OO {z^{-1} }$

Taking the derivative $z$:


 * $(1): \quad \dfrac {\map {\Gamma'} z} {\map \Gamma z} = \ln z - \dfrac 1 {2 z} + \dfrac \d {\d z} \map \OO {z^{-1} }$

Since there exists $\map c \epsilon > 0$ such that:


 * $\forall \size z > 1: -\dfrac c {\size {z^{-1} } } < \size {\map \OO {z^{-1} } } < \dfrac c {\size {z^{-1} } }$

it follows directly that the third term in $(1)$ is $\map \OO {z^{-1} }$.