About a reference collecting the main properties of the modulus and the argument of $f$

Question

Let $f$ be an analytic function in the whole complex plane. We can write $f$ in its polar form: $$f(z)=ρ(z)exp(iθ(z))$$ My question is about a reference collecting the main properties of the modulus and the argument of $f$.

Answer 1

Actually, it is not clear how to write $f$ is polar form globally. The function $\theta$ is undefined at the zeros of $f$, and does not have a single-values continuous branch around them. The most basic properties of these functions are:

• $\rho$ is subharmonic on $\mathbb C$, and harmonic on $\mathbb C\setminus f^{-1}(0)$.
• $\theta$ is harmonic on simply-connected subdomains of $\mathbb C\setminus f^{-1}(0)$.

As for references:

• Potential Theory in the Complex Plane by Ransford covers the fundamental properties of harmonic and subharmonic functions, relating them to holomorphic functions.
• Any book on entire functions considers the global behavior of $\rho$ in detail (sometimes, in excruciating detail). For example, entire functions by R.P. Boas or Introduction to the theory of entire functions by A.S.B. Holland. For the devotees of the subject, Subharmonic functions Vol II by Hayman can be recommended, but this is definitely not a first reading. I think these books don't spend much time on $\theta$ because it's not a globally defined function.
• There is also Harmonic function theory (Axler, Bourdon, Wade) if you are specifically interested in harmonic functions.

As far as I know, nobody ever wrote a book to "collect the main properties of the modulus and the argument"; generally speaking, this is not what mathematicians do.