Let $f:ℂ→ℂ$ be an analytic function. Define $|f|$ and $\arg(f)$ be the modulus and the argument of $f$. Generally, $|f|$ and $\arg(f)$ are not analytic.
My question is about the cases where this happen: $|f|$ and $\arg(f)$ are analytic.
Let $f:ℂ→ℂ$ be an analytic function. Define $|f|$ and $\arg(f)$ be the modulus and the argument of $f$. Generally, $|f|$ and $\arg(f)$ are not analytic.
My question is about the cases where this happen: $|f|$ and $\arg(f)$ are analytic.
The values of a nonconstant analytic function defined on some domain $\Omega\subset{\mathbb C}$ always fill another two-dimensional domain $\Omega'\subset{\mathbb C}$. Given an analytic $f:\ \Omega\to{\mathbb C}$ the two functions $|f|$ and $\arg(f)$ are real-valued, resp. $S^1$-valued; therefore they cannot be analytic.
There is, however, a way to introduce these two functions into the analytic realm: Assume that $\Omega$ is simply connected and that $f(z)\ne0$ for all $z\in\Omega$. Then there is an analytic function $z\mapsto g(z)$ such that $f(z)=e^{g(z)}\quad (z\in\Omega)\ .$ This function $g$ can be considered as the logarithm of $f$. Going through the details you will see that in fact $g(z)=\log\bigl|f(z)\bigr| + i \arg\bigl(f(z)\bigr)\quad (z\in\Omega)$ (note that $g$ is only determined up to an additive multiple of $2\pi i$). So you can say that $\arg\bigl(f(z)\bigr)$ is the imaginary part of $\log\bigl(f(z)\bigr)$.