Characterization of Cosine Integral Function

Definition
Let $\Ci: \R_{>0}: \R$ denote the cosine integral function:


 * $\map \Ci x = \displaystyle \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {\cos t} t \rd t$

Then:


 * $\map \Ci x = -\gamma - \ln x + \displaystyle \int_{t \mathop \to 0}^{t \mathop = x} \frac{1 - \cos t} t \rd t$

where $\gamma$ is the Euler-Macheroni constant.