# Questions tagged [specific-calculation]

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

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### 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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### 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 $$A(x_1,..,x_k):=\{(y_1,\dots,y_k)\in [0,1]^k: \text{there exist intervals }I_1,\dots,I_k \text{ in }[0,1]$$ ...
132 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 ...
189 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 ...
326 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(...
This is a bit of recreational integration. The following, rather attractive integral is quite straightforward via residues: $$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$ ...
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(\...