Characterization of Cosine Integral Function

Definition
Let $\operatorname{Ci}$ denote the cosine integral function:


 * $\operatorname{Ci}: \left({0 \,. \,. \,\to}\right] \to \R$:


 * $\operatorname{Ci} \left({x}\right) = -\displaystyle \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {\cos t} t \, \mathrm d t$

Then:


 * $\operatorname{Ci}\left({x}\right) = \gamma + \ln x + \displaystyle \int_{t \mathop \to 0}^{t \mathop = x} \frac{\cos t - 1} t \, \mathrm dt$

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