# Questions tagged [specific-calculation]

Every question that deals with calculating the value of a specific integral, derivative, summation etc.

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### Fast checking that a system of polynomial equations is satisfiable over $\mathbb{F}_2$

I have a (fairly large) system of polynomial equations, of the form $$c_1d_1=0,\ c_1d_2+c_2d_1=0,\ldots$$ (In case it is relevant, all the polynomials are homogeneous of degree 2, except for exactly ...
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1 vote
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### Computational comparison in solving two optimization problems

Can I get some inputs on whether the following two optimization problems are computationally the same, or one of the problems is easier to solve computationally than the other, such as, finding their ...
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### Closed form of integral $\mathcal{P}\int_{-\infty}^{+\infty}dy\frac{ \text{Li}_{k}(\exp(-(y-\frac{\xi }{2})^2))}{\exp (2 \xi y)-1}$

How could one calculate the closed form solution of this integral: $\mathcal{P}\int_{-\infty}^{+\infty}dy\frac{ \text{Li}_{k}(\exp(-(y-\frac{\xi }{2})^2))}{\exp (2 \xi y)-1}$ Here the integral is ...
• 71
199 views

1 vote
596 views

### Time ordered integral involving beta function:

Any help on unpacking integrals of the following type, would be helpful: $$\int_0^1 \int_0^r r^a (1-r)^b t^n (1-t)^m dr dt$$ where $a, b, n, m \in \mathbb{N}$ and $0 \le t \le 1$. Edit/...
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225 views

### lower bound volume of a set

Let $\lambda$ be Lebesgue measure on [0,1]. For any $x_{1},\dots,x_{k}$ in $[0,1]$, define $$A(x_1,..,x_k):=\{(y_1,\dots,y_k)\in [0,1]^k: \text{there exist intervals }I_1,\dots,I_k \text{ in }[0,1]$$ ...
141 views

### Numerical Calculations

i have this numerical calculation problem : $$\prod \limits_{i=121443371}^{455052511} 1+\frac{1}{p(i)} \leq 1.06406506887043952285362856325019948$$ such that $p(i)$ is the $i$-th prime number i ...
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### Rotation invariance of an integral

Consider the integral depending on 2 parameters $$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$ where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...
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256 views

### Analytic continuation of a specific integral with respect to a parameter

The following integral absolutely converges for $Re(z)<0$ and is analytic in this domain: $$F(z):=\int_{0}^{+\infty}\frac{\sin r}{r}e^{\sqrt{r^2+m^2}z}dr ,$$ where $m>0$ is fixed. Question. To ...
• 21.2k
340 views

### Two similar integrals

Let $n$ be a given even positive integer. We have the following integral \begin{align} \int_0^{\infty}\cdots\int_0^{\infty}e^{-(x_1+\cdots+x_n+y_1+\cdots+y_n)}\prod\limits_{i=1}^n\prod\limits_{j=1}^n(...
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### Evaluating an integral using real methods

This is a bit of recreational integration. The following, rather attractive integral is quite straightforward via residues: $$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$ ...
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Consider the function $$f(x_1,x_2)=|x_1x_2|^{-\alpha/2}\int_{\mathbb{R}} \frac{e^{it(x_1+u)}-1}{i(x_1+u)} \frac{e^{it(x_2-u)}-1}{i(x_2-u)} |u|^{-\beta}du.$$ It is known that \$f(x_1,x_2)\in L^2(\...