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Contour integraion of a rational function of logarithm and exponential
$$\int_{0}^\infty \frac{\log(x)\cos(x)}{(x^2+1)^2}\,dx = – \frac{\pi \mathrm{Ei}(1)}{4e}\frac{\pi}{4e}$$ $$\textit{proof}$$ Consider the following function $$f(z) = \frac{\log(z) }{(z^2+1)^2}e^{iz}$$ Now consider the principle logarithm where $$\log(z) = \logr+i \theta \,\,\, , \theta \,\in (\pi , \pi]$$ Consider the following contour Then by … Continue reading
Posted in Contour Integration
Tagged contour, cosine, exponential, fraction, Integral, proof, residue, theorem
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