Characterization of Cosine Integral Function

Definition

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

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

Then:

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

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

Proof

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Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Miscellaneous Special Functions: Cosine Integral: $35.14$

• Weisstein, Eric W. "Cosine Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CosineIntegral.html