Why do we care about Cauchy principal value?

Question

Question is in the title, basically. I don't understand the motivation behind assigning the Cauchy principal value to otherwise divergent integrals. I'm more comfortable with things like Abel summation that assign values to divergent series, because in my (limited) experience, all the reasonable ways you can do this lead to the same values. But Cauchy p.v. isn't compatible with substitutions, which are normally one of the fundamental tools for evaluating integrals. So why do we care about Cauchy principal value?

Answer 1

The Cauchy principal value of the "integrals" $f\to \int {f(x)\over x}\,dx$ is (uniquely) characterizable in various ways. These are not so much related to complex analysis, as to functional analysis, although certainly an understanding of complex analysis is necessary for some aspects.

For one, up to constant multiples, it is the unique distribution (=continuous linear functional on test functions) that is odd, and is homogeneous of degree $-1$ (suitably normalized).

Up to a constant, it is the Fourier transform of the (tempered) distribution given by integration against the sign function.

Up to a constant, it is the analytic continuation of the distribution-valued function $s\to x/|x|^s$ to $s=2$.

Up to a constant, it is the unique odd distribution $u$ such that $xu=1$. (Note that $x\delta=0$, but/and $\delta$ is even.)