# Tag Archives: contour

## Euler reflection formula proof using contour integration

$$\int_{0}^{\infty}\frac{x^{\alpha}}{x+1}\,dx=- \pi \csc(\pi \alpha)$$ $$\textit{proof}$$ Consider the following function $$f(z) = \frac{z^{\alpha}}{1+z} = \frac{e^{\alpha \log(z)}}{1+z}$$ As we know the function $\log(z)$ is multi-valued defined as $$\log(z) = \ln|z|+i\theta +2k\pi i$$ This maps the complex plain more than once … Continue reading

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Prove that $$\int^{1}_{0} \sqrt{x}\sqrt{1-x}\,dx = \frac{\pi}{8}$$ $$\textit{proof}$$ Consider the function $$f(z) = \sqrt{z-z^2} = e^{\frac{1}{2}\log(z-z^2)}$$ Consider the branch cut on the x-axis $$x(1-x)\geq 0\,\, \implies \, 0\leq x \leq 1$$ Consider $w= z-z^2$ then $$\log(w) = \log|w|+i\theta,\,\, … Continue reading Posted in Contour Integration | Tagged , , | Leave a comment ## Contour integraion of a rational function of logarithm and exponential$$\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) = \log|r|+i \theta \,\,\, , \theta \,\in (-\pi , \pi]$$Consider the following contour Then by … Continue reading Posted in Contour Integration | | Leave a comment ## Bromwich contour integration of the gamma function$$\frac{1}{2\pi i}\int^{c+i\infty}_{c-i\infty}\Gamma(a+t)\Gamma(b-t) s^{-t}\,dt= \frac{\Gamma(a+b)}{(1+s)^{a+b}}s^a\textit{proof}$$Consider the following function$$f(z) = \Gamma(z+a)\Gamma(b-z) s^{-z} Suppose that $a,b \in \mathbb{R}$ and $a < b$. Note that the Gamma function has a pole of order 1 at each non-positive integer where … Continue reading

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