Euler-Mascheroni Constant as Difference of Integrals involving Cosine

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Theorem

$\displaystyle \int_0^1 \frac {1 - \cos x} x \rd x - \int_1^\infty \frac {\cos x} x \rd x = \gamma$

where $\gamma$ is the Euler-Mascheroni Constant.


Proof

From Characterization of Cosine Integral Function, we have:

$\displaystyle \int_t^\infty \frac {\cos x} x \rd x = -\gamma - \ln t + \int_0^t \frac {1 - \cos x} x \rd x$

for all real $t > 0$.

Rearranging:

$\displaystyle \int_0^t \frac {1 - \cos x} x \rd x - \int_t^\infty \frac {\cos x} x \rd x = \gamma + \ln t$

Setting $t = 1$ gives the result, from Natural Logarithm of 1 is 0.

$\blacksquare$


Sources