It's known that for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ the set of points of discontinuity must be an $F_{\sigma}$. In the book "Understanding Analysis" by Abbott is …

What is the principal value of the integral $$\int \limits _0^\infty \left( \frac {1}{x^2}-\frac{\cot(x)}{x} \right) dx ?$$ Maple finds $PV \int_0^\infty \tan(x)/x dx = \pi/2.$ Su …

An old problem: to show that every bounded left integrable function is also Riemann integrable. I know some (for me) not elementary proofs: Gillespie and Kristensen et al (theorem …

Does there exist any rational $t$, such that $|t|<1/10$ and the integral $$\frac{1}{\pi^2}\int\limits_0^{\pi t}\arccos{\left(\frac{\sin{s}}{1+2\sin{s}}\right)}ds$$ is irrational …

Consider a function f continuous on a compact interval. Approximate it by a sequence of polygonal functions (you can). Then consider a sequence of primitives of the polygonal fun …

AMM problem 11621 asks to calculate the integral $$I_2=\int\limits_{-\infty}^{\infty}ds_1\int\limits_{-\infty}^{s_1}ds_2 \int\limits_{-\infty}^{s_2}ds_3\int\limits_{-\infty}^{s_3} …

It can be proved by induction that the recurrence relation $$K_n(u,v)=\int_0^1K_1(u,u_1)K_{n-1}(u_1,v)du_1,$$ with $K_1(u,v)=\theta(1-u-v)$, where $\theta(x)$ is the Heaviside step …

I need the coefficients of the Fourier series expansion of $\tanh(\alpha cos(x))$, for every real positive number $\alpha >0 $ namely, since it is an even function, I need the comp …

While studying a problem about orthogonal polynomials I encountered the following expressions \begin{equation} f(n)=\sum_{k=0}^{n}(-1)^k\binom{n+k}{2k} \frac{1}{k+1}\binom{2k}{k} …

Hello all, I have happened upon the following sum: $ 1^2 + \Big(1 \times \frac{1}{3} + \frac{1}{3} \times 1 \Big)^2 + \Big(1 \times \frac{1}{5} + \frac{1}{3} \times \frac{1}{3} …

An important ingredient in recent progress on Euclidean harmonic analysis has been that of "inductions on scales". A few examples are the papers of Wolff, Tao, and Bourgain and Gut …

Hi, Let $\Lambda\subset\mathbb{R}$ be an infinite discrete set of finite density (for simplicity one may take the density equals 1) and $\delta_{\lambda}$ is a unit mass located at …