Bachelor thesis Non-absolute convergence of Newton integral (CZ/ENG)

In this thesis we search for sufficient and necessary conditions for non absolute convergence of Newton integral of function of the form sin(φ(x))/x. Importantly we analyse how the oscilation of the sine function influences the convergence of the integral. We are dealing with continous non-decreasing functions such that limx→∞ φ(x) = ∞. We proved that bilipschitz of φ is not sufficient. Nevertheless, we proved several theorems about sufficient conditions for the convergence of the integral. I have proven following theorem.

theorem - relationship between periodicity and convergence of newton integral

Let \(\psi \) be a function that is \( \underline{2p-periodic} \) and \( \underline{continous} \) on \( \mathbb{R} \), then

\[ \begin{equation*} \int_{1}^{\infty} \frac{\psi(x)}{x} \,dx\, \, \text{convergence} \Leftrightarrow \int_{-p}^{p} \psi(x)\,dx\, =0. \end{equation*} \]