Extension of Harmonic Number to Non-Integer Argument

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

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:
 * $\forall n \in \N: H \left({n}\right) = H_n$

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