I am trying to show that assuming for an analytic $f$ we have $|f|$ is also analytic. That the original f must be constant.
My original thought was to use the Cauchy Riemann equations to try and show that the partial derivatives are equal to their negatives. I am unsure how to carry this out however, as I am struggling to express the partial derivatives of $|f|$ in terms of the partial derivatives of $f$.
Could someone please let me know if I'm on the right track, or else give me a nudge in the correct direction.
Thanks in advance
Best Answer
If $f$ and $|f|$ are both analytic (in the whole complex plane $\mathbb C$), $g:=\frac{f}{1+|f|}$ is analytic in the whole complex plane and $|g|< 1$. So $g$ is constant $f=c(1+|f|)$ and, in particular, $|f|=|c|(1+|f|)=|c|+|c||f|$ and $|f|=\frac{|c|}{1-|c|}$ and you have that $|f|$ is constant. Then, using that $f=c(1+|f|)$, you'll get that $f=c\left(1+\frac{|c|}{1-|c|}\right)$ which is a constant.