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 Post subject: Trigonometric logarithmic integralPosted: Tue May 23, 2017 2:30 pm

Joined: Sat Nov 14, 2015 6:32 am
Posts: 159
Location: Melbourne, Australia
Let $0 \leq \alpha, \beta \leq \pi$ and $\kappa>0$. Prove that

$\int_0^{\infty} \frac{1}{x} \log \left( \frac{x^2 + 2\kappa x \cos \beta + \kappa^2}{x^2 + 2 \kappa x \cos \alpha + \kappa^2} \right) \, {\rm d}x = \alpha^2 - \beta^2$

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$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$

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