Characterization of Cosine Integral Function

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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.


Proof


Sources