# Tagged Questions

Questions on the evaluation of integrals along a locus in the complex plane.

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### Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
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### Integral with contours

I want to evaluate the integral $\displaystyle \int_0^\infty \dfrac{\ln x}{e^x+1}\,{\rm d} x$ using contour integration. At first I though using a rectangular. Problem is that I cannot establish the ...
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### Solving an integral (using Cauchy contour integral?)

I need to solve this integral: $$f(t)=\int_0^\infty x^2 \sqrt x \left( e^{a x} -1\right)^{-1/2} \frac{e^{i(b-x)t}-1}{b-x} dx$$ where $a$ and $b$ are real, positive ...
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### Evaluate $\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$ using residue calculus

I'm asked to evaluate $$\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$$ $\mathbb{R}\ni a>0$, using residue calculus (where $\sqrt{\cdot}$ is the PV $\sqrt{}$). My approach is as follows: ...
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### An interesting identity involving powers of $\pi$ and values of $\eta(s)$

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
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### Proof of Sophomore's Dream using Contour Integration

Sophomore's dream is a relatively common identity, that states $$\int _0^1 x^{-x} dx = \sum_{n = 1}^\infty n^{-n}$$ The common proof is found using the series expansion for $e^{- x \log x}$ and ...
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### Showing that $\lim_{N \to \infty} \int_{C_{N}} \frac{ \sinh az}{\sinh \pi z} \mathrm{e}^{ibz} \, dz =0$

One of several ways to evaluate $$\int_{0}^{\infty} \frac{\sinh ax}{\sinh \pi x} \cos (bx) \, dx \ , \, |a|< \pi$$ is to sum the residues of $f(z) = \frac{\sinh az}{\sinh \pi z} \mathrm{e}^{ibz}$ ...
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### Satisfying a Differential Equation and complex Laguerre

I have the following problem Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
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