I believe that since $|x|^2=x^2$ then we have the Fourier transforms
$\int_{-\infty}^{\infty} \mathrm dx \frac{\exp{iux}}{a^2+|x|^2} =\int_{-\infty}^{\infty}\mathrm dx \frac{\exp{iux}}{a^2+x^2}$
even if the function $|x|$ is not analytic at $x=0$. Also, if we are summing over complex numbers the two sums $\sum_n (c_n)^2$ and $\sum_n |c_n|^2$ are not equal if the $c_n$ are complex numbers.