# Monthly Archives: May 2017

## Integrating a cosine log integral around a semi-circle contour

Prove that  $$\int^1_0 \frac{\cos(\log x)}{x^2+1}\,dx = \frac{\pi}{4}\mathrm{sech}\left( \frac{\pi}{2}\right)$$ First note that $$2 \int^1_0 \frac{\cos(\log x) }{x^2+1}\,dx = \int^\infty_0 \frac{\cos(\log x)}{x^2+1}\,dx$$ Integrate the following function $$f(z) = \frac{e^{i\log(z)}}{z^2+1}$$ Around a semi-circle in the upper half place. Where we avoid the branch … Continue reading

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## Creating Difficult integrals by the residue theorem

Theorem Let $$f$$ be analytic function in the unit circle $$|z|\leq 1$$  such that $$f\neq 0$$ . Then $$\int^{2\pi}_0f(e^{it})\,dt =2\pi \, f(0)$$ $$\textit{proof}$$ Since the function $$f$$ is analytic in and on the … Continue reading