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If $f:U\subset\mathbb{C}\mapsto\mathbb{C}$, where $f(x+iy)=u(x,y)+iv(x,y)$ is a meromorphic function and if $f$, $f'$, and $f''$ are not zero in the strip $a<x<b$, can we get $\frac{\partial}{\partial x}|f|^2$ is not zero?


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up vote 2 down vote accepted

Try $f(z) = e^{iz}$, noting that $|f(z)| = e^{-y}$

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