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A logarithmic Poisson integral

Let $a \geq 0$. We will prove that

$$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, \mathrm{d}x = \left\{\begin{matrix}
0 & , & \left | a \right | \leq 1 \\
2 \pi \ln \left | a \right | &, & \text{otherwise}
\end{matrix}\right.$$

Background:...




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Pythagoras' theorem 3D version

The one theorem that almost anyone who has ever studied Mathematics remembers is the Pythagoras’ theorem. It is the most well-known theorem for about 2,500 years now and has allowed the mankind to evolve in a myriad of ways. Its applications are limitless and range from construction to even...

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