Is modulus of a complex function always non holomorphic?

Question

Let $f(z)$ be a complex function that is holomorphic on an open subset of the complex plane. Now, if we define another function $g(z)=|f(z)|^2$, can we say anything about the holomorphicity of $g(z)$ for arbitrary $f(z)$? Also, can you give examples where depending on the form of $f(z)$, the function $g(z)$ turns out to be holomorphic or otherwise?

Note: I am not a mathematician, so there may be mistakes in the statement of my problem. Please do point that out.

@Did The form of $f(z)$ is arbitrary, so can't use Cauchy Riemann equations. In that case is it possible to say whether $g(z)$ is analytic? — damaihati, Sep 17 '18 at 14:34

score 2 · Accepted Answer · answered Sep 17 '18 at 14:52

2

Note that $g(z)=|f(z)|^2$ is a real valued function.

That is you have a function $u+iv$ for which $v=0$

Cauchy- Riemann theorem could be applied to make a decision regarding the function $g(z)$ being or not being holomorphic.

answered Sep 17 '18 at 14:52

Mohammad Riazi-Kermani

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Is modulus of a complex function always non holomorphic?

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