# Tagged Questions

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### How to prove $\int_1^\infty\frac{K(x)^2}x dx=\frac{i\,\pi^3}8$?

How can I prove the following identity? $$\int_1^\infty\frac{K(x)^2}x dx\stackrel{\color{#B0B0B0}?}=\frac{i\,\pi^3}8,\tag1$$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind: ...
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### Quarternionic Analysis

What is/are the current understanding/opinions about Quarternionic Analysis as a generalization of Complex Analysis with respect to a "Quarternionic Residue Calculus" (if such a thing exists)? i.e. ...
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### Integrating $\int_0^\infty\frac{\log (1+\frac{z^2}{4\pi^2})}{\operatorname e^z-1}\, \operatorname d\!z$ using residue calculus.

I've been looking at how to integrate the following definite integral using the residue calculus, but can't seem to get my thoughts together. I've checked the integrand satisfies the Cauchy Riemann ...
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### integral to infinity + imaginary constant

A proof I'm reading tries to evaluate the integral (where $i$ is the regular imaginary unit) $$\int_{-\infty}^{\infty} e^{-(x-\alpha i)^2}\mathrm{d}x$$ by doing a substitution $u=x-\alpha i$. ...
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### Calculating $\int_0^\infty \frac {\sin^2x}{x^2}dx$ using the Residue Theorem.

I am trying to compute the following integral using the Residue Theorem but am quite stuck: $$\int_0^\infty \frac{\sin^2x}{x^2}dx$$ I have tried applying Jordan's lemma, having written $\sin(x)$ as ...
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### Simpler way to evaluate the Fourier transform of $\exp\left(i e^x\right)$?

I have the task to evaluate $|a(k)|^2$ with $$a(k) = \int_{-\infty}^\infty \!dx\,\exp\left(i k x + i e^{x}\right).\tag{1}$$ The integral in (1) can be evaluated explicitly via the substitution ...
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### Can we calculate this integral using the Residue Theorem?

Calculate this integral using the Residue Theorem: $$\int_0^{\infty}\frac {\text{d}x}{x^5 + 1}$$
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### Evaluation of a definite integral

I want to find the best way to show $\int_0^\infty\dfrac{x^{2m}}{x^{2n}+1}\,dx=\dfrac{\pi}{2n}\operatorname{csc}\left(\dfrac{2m+1}{2n}\pi\right)$, where $0\leq m<n$. It's easy to verify some ...
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### Integral $\int_{0}^{1}\frac{1}{x^{2}+2x+2}dx$ via contour integration

I want to evaluate the following integral $$\int_{0}^{1}\frac{1}{x^{2}+2x+2}dx$$ by contour integration; I have a problem with the choice of the contour/ branch cuts. Where can I find some some ...
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### Help with a contour integration

I've been trying to derive the following formula $$\int_\mathbb{R} \! \frac{y \, dt}{|1 + (x + iy)t|^2} = \pi$$ for all $x \in \mathbb{R}, y > 0$. I was thinking that the residue formula is the ...
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### Can we integrate $\int_a^b t^i f(t) \, dt$

If we are given reals $a$ and $b$, and we have a function of $t$, $f(t)$, we can analyze the integral: $$\int_a^b t^i f(t) \, dt$$ ...where $i$ is the imaginary number. I'm wondering if we can ...
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### Behavior of $f(z)=\int_0^1\mathrm{e}^{\alpha t^2}\sin(tz)\,dt$ when $\alpha <0$

Define $$f(z)=\int_0^1\mathrm{e}^{\alpha t^2}\sin(tz)\,dt,$$ where $\alpha \in \mathbb{R}$. If $\alpha >0$ then $f(z)$ has infinitely many real zeros and at most a finite number of complex ...
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### Probably Riemann surface integral

Here is the integral: May you please suggest some beautiful idea on using Riemann surface, or some Gauss-Ostrogradsky at the beginning. Also, the initial integral looks really symmetric, so maybe ...
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### Is $f(z)=\int_{-\infty}^{\infty}c(k)e^{-k^2/2}e^{ikz}dk$ a general analytic function?

I have an expression $f(z)=\int_{-\infty}^{\infty}c(k)e^{-k^2/2}e^{ikz}dk$ where $c(k)\in\boldsymbol{C}$ and $k\in\boldsymbol{R}$. $f(z)$ is an analytic function, since it contains only non-negative ...