Modulus and argument of a holomorphic function.

Question

It's well known that $\Re(z)$ and $\Im(z)$ are harmonic functions (where $z$ is a holomorphic function). What about the modulus and the argument?

Answer 1

With the reservation stated in the other answer, the different branches of the argument are harmonic in subsets strictly lesser than the plane. For example: $$\arg(x + yi) = \arctan(y/x)\hbox{ (main branch of $\arctan$)}$$ is harmonic: $$ \frac{\partial^2}{\partial x^2}\arctan(y/x) + \frac{\partial^2}{\partial y^2}\arctan(y/x) = \frac{2xy}{(x^2 + y^2)^2} - \frac{2xy}{(x^2 + y^2)^2} =0. $$ Modulus isn't harmonic, but its $\log$ is harmonic in $\Bbb C\setminus\{0\}$: $$\frac{\partial^2}{\partial x^2}\log\left(\sqrt{x^2 + y^2}\right) + \frac{\partial^2}{\partial y^2}\log\left(\sqrt{x^2 + y^2}\right) = \frac12\left(\frac{y^2 - x^2}{(x^2 + y^2)^2}\right) + \left(\frac{x^2 - y^2}{(x^2 + y^2)^2}\right) = 0. $$

Answer 2

No and no.

For the modulus, note that if $f(z)=z^2$, then $\Delta |z|^2 = 4 \neq 0$.

For the argument, the situation is even worse: there is no continuous choice of argument for $f$ (that is, there is no continuous function $\theta : \mathbb C \to \mathbb R$ such that $f(z)=|z|^2 e^{i \theta(z)}$. Since harmonic functions are continuous, the argument cannot be harmonic.