Sep 4, 2013 · How do I calculate a Cauchy random variable and how do I calculate the probability mass function to show it is "heavy tailed"

My question is related with the definition of Cauchy sequence As we know that a sequence $(x_n)$ of real numbers is called Cauchy, if for every positive real number ε, there is a positive …

Also a few other equations related to this equation are often studied. (Equations which can be easily transformed to Cauchy functional equation or can be solved by using similar methods.) …

I have shown an example of how to use the definition of a Cauchy sequence. I changed the sequence to an easier one (to be honest because the one you suggested looked like a mess). …

We need to prove that differentiation under the integral is permissible. To do so, we will estimate the difference between the difference quotient, f(a+Δz)−f(a) Δz,and the result we obtain from …

Sep 30, 2017 · A purpose of the Cauchy principal value is to rectify this problem, to take into account oscillations like the Riemann integral does and give a meaningful number that …

You know that every convergent sequence is a Cauchy sequence (it is immediate regarding to the definitions of both a Cauchy sequence and a convergent sequence). Your demonstration of …

Cauchy's Formula has a remarkable interpretation in terms of hyperbolic geometry. To understand it, you need to know very little about hyperbolic geometry.

We define γw: = γ1 + ⋯ + γ5 and then our integral is zero (Cauchy's integral theorem). On the other hand, we can calculate the several integrals separately with (let R → ∞ and ε → 0.)

Very good proof. Indeed, if a sequence is convergent, then it is Cauchy (it can't be not Cauchy, you have just proved that!). However, the converse is not true: A space where all Cauchy …