Let $f : (-\infty, \infty) \rightarrow \mathbb{C}$ be an even ,smooth, and compactly supported, vanishing at zero.
Is there a way to solve integral
$\int_0^\infty f'(x)/x \; d x$
in a suitable interpretation a la integration by parts.
Let $f : (-\infty, \infty) \rightarrow \mathbb{C}$ be an even ,smooth, and compactly supported, vanishing at zero.
Is there a way to solve integral
$\int_0^\infty f'(x)/x \; d x$
in a suitable interpretation a la integration by parts.
The question in the title and the question in the text are different. I assume the one in the text is the correct one. In that case, let $\varepsilon > 0$. We get (by partial integration)
$ \int_{\varepsilon}^\infty \frac{f'(x)}x\,dx = \left[ \frac{f(x)}x \right]_\varepsilon^\infty + \int_{\varepsilon}^\infty \frac{f(x)}{x^2}\,dx.$
Let $\varepsilon \searrow 0$. Then the first term tends to $-f'(0)$ (the value at the upper bound is $0$ since $f$ is compactly supported). The second term could very well diverge though. Take for example $f(x) = x\psi(x)$ where $\psi$ is a cutoff function that is equal to $1$ on a neighborhood of $0$.