# Tag Archives: contour

## 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

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Prove for $a,b > 0$ $$\int^{2\pi}_0\frac{dt}{a^2\cos^2 t +b^2\sin^2 t} = \frac{2\pi}{ab}$$ $$\textit{solution}$$ Let us integrate the following function $$f(z) = \frac{1}{z}$$ Around the ellipse $$\oint_{\gamma}f(z)\,dz =2\pi i\,\mathrm{Res}(f,0)$$ The parametrization of the ellipse $\gamma(t) = a\cos(t)+ib\sin(t)$ $$\oint_{\gamma}f(z)\,dz=\int^{2\pi}_0 \frac{-a\sin t+ib\cos t}{a\cos t+ib\sin t} … Continue reading Posted in Contour Integration | | Leave a comment ## Integrating a fraction of exponential and trignometric using rectangular contour [Ex 41 ] Watson’s complex integration$$\int^\infty_0 \frac{\sin(ax)}{e^{2\pi x}-1}\,dx = \frac{1}{4}\coth\left(\frac{a}{2} \right)-\frac{1}{2a}\textit{solution}$$By integrating the following function$$f(z) = \frac{e^{iaz}}{e^{2\pi z}-1}$$The function is analytic in and on the contour, indented at the poles of the function Hence by … Continue reading Posted in Contour Integration | | Leave a comment ## Integrating a function around three branches using a semi-circle contour [Ex] Watson’s complex integration$$\int^{\pi/2}_{0}\cos(nt)\cos^m(t)\,dt=\frac{\pi \Gamma(m+1)}{2^{m+1}\Gamma\left(\frac{n+m+2}{2}\right)\Gamma\left(\frac{2-n+m}{2}\right)}\textit {solution}$$Let us integrate the following function$$f(z) = z^{n-m-1}\left(1+z^2\right)^m$$We choose the principle logarithm where$$\log(z) = \log|z|+\mathrm{Arg}(z)$$Note that the function $z^{n-m-1} = e^{(n-m-1)\log(z)}$ will have a branch cut on the … Continue reading Posted in Beta function, Contour Integration | | Leave a comment ## Integration related to gamma function using rectangle contour [Ex 9] Watson’s complex integration$$\int^{\infty}_{-\infty}e^{-x^2}\,\cos(2ax)dx=e^{-a^2}\sqrt{\pi}proof$$Integrate the following function$$f(z) = e^{-z^2}$$Use the following contour Note that the function is entire, hence$$\int^{L}_{-L} e^{-t^2}\,dt+\int^{L+ai}_{L} e^{-t^2}\,dt+\int^{-L}_{-L+ai} e^{-t^2}\,dt+\int^{-L+ai}_{L+ai}e^{-t^2}\,dt=0$$For the forth integral use the substitution $x= t-ai$$$\int^{-L+ai}_{L+ai}e^{-t^2}\,dt=\int^{-L}_{L}e^{-(x+ai)^2}\,dx=-e^{a^2}\int^{L}_{-L}e^{-x^2}\,e^{-2iax}dx$$Take … Continue reading Posted in Contour Integration, Gamma function | | Leave a comment ## Integral of arctan and log using contour integration$$\int^\infty_0\frac{\log\left(x^2+1 \right)\arctan^2\left(x\right)}{x^2}\,dx = \frac{\pi^3}{12}+\pi \log^2(2)$$Lemma$$\int^\infty_0 \frac{\log^3(1 + x^2)}{x^2}\,dx = \pi^3+ 3 \pi \log^2(4)$$Start by the following$$\int^{\infty}_0 x^{-p}(1+x)^{s-1} dx= \frac{\Gamma(1-p)\Gamma(p-s)}{\Gamma(1-s)}$$Let $x\to x^2$$$\int^{\infty}_0 x^{-2p+1}(1+x^2)^{s-1} dx= \frac{\Gamma(1-p)\Gamma(p-s)}{2\Gamma(1-s)}$$Let $p = 3/2$$$\int^{\infty}_0 \frac{1}{x^2(1+x^2)^{1-s}} … Continue reading

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$$\int^{\infty}_{-\infty} \frac{\cos(ax)}{\cosh(x)} \,dx = \pi \, \mathrm{sech} \left( \frac{\pi a}{2}\right)$$ $$\textit{proof}$$ Consider $$f(z) = \frac{e^{iaz}}{\sinh(z)}$$ If we integrate around a contour of height $\pi$ and stretch it to infinity we get By taking $T \to \infty … Continue reading Posted in Contour Integration | | Leave a comment ## Integrating along the unit circle Prove that \int^{2\pi}_0e^{\cos \theta}\cos(n\theta -\sin \theta)\,d \theta=\frac{2\pi}{n!} \textit{proof} Consider the following function f(z)=e^{z^{-1}}z^{n-1} Now we integrate the function along a circle of radius 1 The contour encloses a pole at \(z = 0$ $$\oint_{|z|=1}e^{z^{-1}}z^{n-1} dz=2\pi i\mathrm{Res}(f(z),0)$$ Now we … Continue reading

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## 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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