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.
Proof
Laplace transform to the one, then invert to the other?
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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. http://mathworld.wolfram.com/CosineIntegral.html
