Question

Using Cauchy-Riemann equations, how do you show that |z|^2 is not analytic?

Answer 1

$|z|^2$ is not an analytic function. It is easy to check that it does not satisfy the Cauchy-Riemann equations anywhere outside of the origin. It is also easy to check that it is not differentiable outside of the origin.

In general, real-valued functions can not be analytic. By some coincidence, they could be differentiable at some points, but they are never analytic (unless they are constant).