Category Archives: Contour Integration

Prove  $a,b,c,d >0$ $$\int^\infty_0 \frac{\log(a^2+b^2x^2)}{c^2+d^2x^2}\,dx = \frac{\pi}{cd} \log \frac{ad+bc}{d}$$ Consider the function $$f(z) = \frac{\log(a-ibz)}{c^2+d^2z^2}$$ We need the logarithm with the branch cut $y<-\frac{a}{b} , x =0$ . Note that this corresponds to $$\log(a+ibz) = \log\sqrt{(a+y)^2+b^2x^2}+i\theta … Continue reading Posted in Contour Integration | | Leave a comment Solving Euler sums using Contour integration Prove that$$\sum_{n=1}^\infty \frac{H_n}{n^2} = 2\zeta(3)\textit{proof}$$Consider the function$$f(z) = \frac{(\psi(-z)+\gamma)^2}{z^2}$$Note that $f$ has poles at non-negative integers By integration around a large circle $|z| = \rho$ Note that$$\oint f(z)\,dz = 2\pi … Continue reading

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Integral of rational function with cosine hyperbolic function using rectangle contour

$$\int^{\infty}_{-\infty} \frac{1}{(5 \pi^2 + 8 \pi y + 16y^2) }\frac{\cosh\left(y+\frac{\pi}{4} \right)}{\cosh^3(y)}dy=\frac{2}{\pi^3}\left(\pi \cosh\left(\frac{\pi}{4} \right)-4\sinh\left( \frac{\pi}{4}\right) \right)$$   $$\textit{proof}$$ Consider $$f(z) = \frac{\sinh(z)}{z \sinh^3(z-\pi/4)}$$ If we integrate around a contour of height $\pi$ and stretch it to infinity we … Continue reading

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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

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