Convolution of a Cauchy distribution and a uniform distribution

Let $X$ have the Standard Cauchy distribution and let $Y$, independently, have the uniform distribution on $[-h,h]$, where $h>0$. The convolution of probability densities gives the probability density of $Z=X+Y$ asThis is an even function. Its derivative isso its unique mode is zero. Its tails satisfy $p_h(z)\sim 1/(\pi z^2)$ as $|z|\to \infty$: multiplying the integral by $z^2$ makes the integrand converge uniformly to $1$ on the bounded interval $[-h,h]$. Consequently both positive and negative parts of the first moment diverge, so its expected value does not exist. The symmetric Cauchy principal value of that moment is zero, which is a different assertion. Adding a bounded independent displacement smooths the probability density without removing the divergent first moment of the Cauchy distribution.