Extension of Harmonic Number to Non-Integer Argument

Theorem
Let $\map H x$ be the real function defined as:
 * $\map H x = \gamma + \dfrac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }$

where:
 * $\gamma$ denotes the Euler-Mascheroni constant
 * $\Gamma$ denotes the gamma function
 * $\Gamma'$ denotes the derivative of the gamma function.

Then $H$ is an extension of the mapping $H: \N \to \Q$ defined as:
 * $\forall n \in \N: \map H n = H_n$

where $H_n$ denotes the $n$th harmonic number.

Proof
For $n \in \N$: