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Recently I met an integral as follow: $$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq i<j\leq9}\sin\frac{\theta_i-\theta_j}{2}\right)\left(\prod\limits_{i=1}^9(1+\cos(\theta_i-\theta_{i+...

Lemma 4.5 of Titchmarsh's book The Theory of the Riemann Zeta function says (slightly rephrased): Let $F$ be a twice differentiable real function such that $ F''(x) \geq r > 0$ for all $x$ in $[a,b]...

Let $k_{j}\in {\mathbb{Z}}^{+}$ and $\,a_{j}\in \,]0,1[$, be such that $k_{j}\,a_{j}<1$, $j=1,\cdots,n$. Let $f:\mathbb{R}^{n}\rightarrow [0,\infty[$ be defined by the integral $$f(y):=\int_{\...

Consider the following oscillatory integral $$ I(n):=\int_{-\pi}^\pi\int_{-\pi}^\pi e^{i n(x+y)}\frac {(1 - \cos(2x)) (1 - \cos(2y))} {2k - (\cos x + \cos y)}\ \mathrm{d}x\,\mathrm{d}y. $$ where $...

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a 1-periodic function. I am looking about the conditions on $(a,b)\in\mathbb{R}^2$ such that we have the property : $$\frac{1}{n}\sum_{\ell=0}^{n-1}f(a+b\...

Hello, When I read Stein's book of Singular Integrals, at p. 118, there is an obvious mistake: $$ \int_{R^n} |x-y|^{-n+\alpha} |y|^{-n+\beta}=\frac{\gamma(\alpha)\gamma(\beta)}{\gamma(\alpha+\beta)},...