Cauchy convolution semigroup ($K_a\ast K_b=K_{a+b},K_a(x)=a/[\pi (x^2+a^2)]$)

The normalized Cauchy distribution densities $K_a$, $a>0$, form a convolution semigroup: convolution adds their scale parameters. This follows from the convolution theorem, since their Fourier transforms are proportional to $e^{-a|k|}$. Consequently an integral equation with a Cauchy kernel can be solved by subtracting scale parameters when the resulting scale is positive.