I found an exercise which asks: Determine whether a function is analytic or not without calculating the Cauchy-Riemann conditions. The functions are
$$ \sin |z|\qquad e^{z^2}\qquad \arg{z^2}\qquad \cos{z^3} $$
So, the first is not analytic, but why? I tried figure out an answer and I thought that $|z| = \sqrt{z\bar{z}}$, giving the function a dependence on $\bar{z}$ which is clearly a not analytic function. $e^{z^2}$ looks analytic as it only depends on $z$, and likewise $\cos{z^3}$.
But what about $\arg{z^2}$ how to know if it is analytic or not?
Thanks