The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Using integration by parts, justify, for $x > 0$, the convergence of the following integral: $$\int_{0}^{+\infty} \frac{h(u)}{u+x} du$$
The function $h$ is defined on $\mathbb{R}$ by
$$h(u) = u - [u] - 1/2$$
Using integration by parts, justify, for $x > 0$, the convergence of the following integral:
$$\int_{0}^{+\infty} \frac{h(u)}{u+x} du$$