I just remember an observation I made a loong time ago, while learning to use Maple.
Let $f_0$ be the cardinal sine. It is well known that:
$$\int_0^{+\infty} f_0 (t) \ dt = \frac{\pi}{2}.$$
Since the indefinite integral converges, we can define:
$$f_1 (x) := - \int_x^{+ \infty} f_0 (t) \ dt,$$
so that $f_1 (0) = -\pi/2$. But let's do this again! For all $n \geq 0$ and all $x \in \mathbb{R}$, let:
$$f_{n+1} (x) := - \int_x^{+ \infty} f_n (t) \ dt.$$
From numerical experiments, it looks like $f_n$ is well-defined, at least for small values of $n$ (until 5 or so). In addition, the sequence $(f_n (0))_{n \geq 0}$ is remarkable : it begins by $(1, -\pi/2, 1, 0, ...)$. I think there were other $\pi$ later in the sequence, but I don't manage to get these terms at the moment. So, a couple of questions:
Is $f_n$ well-defined for all $n$?
If so, what is the sequence $(f_n (0))_{n \geq 0}$?