Lets define new differintegral formula as $$\mathbb{D}^s_xf(x)= \sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ or, equivalently, $$\mathbb{D}^s_xf(x …

Mamikon's visual calculus (see Mamikon, Tom Apostol, Wikipedia) is a very beautiful and surprisingly efficient tool. The basis is Mamikon's theorem. The area of a tangent swee …

Let $u(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Suppose that for some real numbers $a,b$ with $a < b$ the following equality is true: \begin{equation} \frac …

This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY. George Berkeley wrote in 1734 with reference to the e …

Let $f$ be a bounded function on a close interval, $[0,1]$ e.g.. Can it be everywhere discontinuous and integrable? Thanks! P.S. It isn't a homework for me and I asked this quest …

Dirac writes down the following formula on page 61 of his "Principles of quantum mechanics": $\frac{d}{dx}\log x = \frac{1}{x} -i\pi\delta(x)$, see http://adsabs.harvard.edu/abs/19 …

Suppose we have a initial segment $x_1,\ldots,x_N$ (for reasonably large $N$) of a sequence of natural numbers $(x_i)$. We have reason to believe the generating function $\sum_{i=0 …

Consider the following gaussian denisity over $\mathbb{R}^{2^n}$ $$p_n(\underline{x}):=\frac{\exp(-\frac{1}{2n} < C_n^{-1}\underline{x},\underline{x} >)}{\sqrt{(2\pi n)^{2^n} \d …

Hi, For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds: $$\sum_{k …

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "o …

In the paper E. Mehlum, Appell and the apple (nonlinear splines in space), Technical Report No. 1676 (1981), Central institute for industrial research, Oslo (reproduced in the book …

Does anyone know of any tight upper/lower bound for incomplete Gamma functions? i.e either of the following functions: $$ \Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t …

Suppose we have functions $g\colon [0,1]\mapsto \mathbb{R}$, which is concave, vanishes at the origin and fullfills condition $$g(xy)/xy\leq g(x)/x+g(y)/y$$ for any $x,y\in[0,1]$ a …

How to take the inverse Mellin of a function having a fractional power lets suppose its f*(s)=(n!)^2/3 ?

Given, $g(Z)=\operatorname{tr}\phi(Z)$, where $\phi(Z)= Z^T\left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right) Z$ where $Z$ is a real rectangular matrix with more rows than co …