Monthly Archives: March 2017

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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Contour integration for a rational function of cos and cosh

$$ \int^{\infty}_{-\infty} \frac{\cos(ax)}{\cosh(x)} \,dx = \pi \, \mathrm{sech} \left( \frac{\pi a}{2}\right)$$ $$\textit{proof}$$ 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

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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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General formula for an integral involving powers of logarithms

$$\int\limits_0^1 \dfrac{\log^{m} (1+x)\log^n x}{x}\; dx = (-1)^{n+1}(n!) (m)! \sum_{\{m-1\}} \sum_{k=1}^\infty \frac{(-1)^k}{k^{n+2}} \prod^{l’}_{j=1}\frac{(-1)^{i_j}}{(i_j)!} \left( \frac{H_{k-1}^{(r_j)}}{r_j}\right)^{i_j}$$ $$\textit{solution}$$ Stirling numbers of the first kind might be useful here, Consider $$m! \sum_{k=m}^\infty (-1)^{k-m} \left[k\atop m\right] \frac{x^k}{k!} = \log^m(1+x)$$ $$\int\limits_0^1 \dfrac{\log^m (1+x)\log^n x}{x}\; dx = … Continue reading

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Multiple integral related to Euler sums

Posted by Cornel Ioan Valean on Facebook Show that $$\small \int^1_0 \cdots \int^1_0 \frac{\mathrm{d}\varphi_1\cdots \mathrm{d}\varphi_5}{(\varphi_1+\varphi_2+\varphi_3+\varphi_4+\varphi_5+1)(\varphi_1+\varphi_2+\varphi_3+\varphi_4+\varphi_5+1)} = \frac{59}{32}\zeta(5)-\frac{1}{2}\zeta(2)\zeta(3)$$ $$\textit{proof}$$ Consider the integral $$I = \int^1_0 \cdots \int^1_0 \frac{\mathrm{d}\varphi_1\cdots \mathrm{d}\varphi_5}{(\varphi_1+\varphi_2+\varphi_3+\varphi_4+\varphi_5+1)(\varphi_1+\varphi_2+\varphi_3+\varphi_4+\varphi_5+1)}$$ Use the series expansion $$I = \sum_{n=0}^\infty \sum_{k=0}^\infty \frac{(-1)^{n+k}}{n+1} \int^1_0\cdots \int^1_0 (\varphi_1\cdots … Continue reading

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