## What $Re(f(z))=c$ can be if $f$ is a holomorphic function ?

Suppose that $f:U\subset\mathbb{C}\to\mathbb{C}$, where $U$ is a region in the complex plane, is a holomorphic function.

Of course, if $c\in\mathbb{R}$ is a regular value for $\text{Re}(f(z))$ then $\text{Re}(f(z))^{-1}(c)$ is an union of differentiable curves in the plane.

Question:

If $c$ is not a regular value and $\text{Re}(f(z))^{-1}(c)$ have at least one cluster point is this set also a piece-wise differentiable curve ?

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Yes, but they may intersect each other. Let us say $z$ is a critical point if $f'(z)=0$, and let us treat the set $\mathrm{Re}f(z)=c$ as the preimage of the curve $\mathrm{Re}w=c$. Then at non-critical points $f$ is locally invertible, so the curve $\mathrm{Re}w=c$ has differentiable curve as its preimage near non-critical points. Now near its critical points $f$ is $n$-to-$1$, and so the curve $\mathrm{Re}w=c$ will have as its preimage $n$ copies of differentiable curves, which all intersect (forming an even angle between them) at the critical point. – timur Nov 26 2010 at 17:17

Yes. The set of points in $\mathbb C$ having real part equal to $c$ form a line, i.e. a smooth simple curve $l$. The counterimage $f^{-1}(l)$ of any smooth simple curve $l$ via a holomorphic function $f$ is always piecewise smooth.

To prove the last sentence, take any point $z_0\in U$ with $f(z_0) \in l$. There are two local diffeomorphisms at $z_0$ and $f(z_0)$ that move $z_0$ to $0$ and transform $f$ locally into $g(z) = z^n$. Since $f$ is not constant, we have $n>0$. Local diffeomorphisms send smooth curves to smooth curves. The counterimage along $g$ of a smooth curve passing through $0$ is the union of $n$ smooth curves exiting from 0. Therefore $f^{-1}(l)$ is a piecewise smooth curve.

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 Hi Bruno, thank you for write this. Nice answer. – Leandro Nov 26 2010 at 23:01

You are essentially asking for a description of the zero-set of a harmonic function in two variables. That is a very delicate issue, but has been studied a lot.

Maybe looking at:

L. De Carli, S. M. Hudson, Geometric remarks on the level curves of harmonic functions, Bull. Lond. Math. Soc. 42 (2010), no. 1, 83–95.

L. Flatto, A theorem on level curves of harmonic functions, J. London Math. Soc. (2) 1 (1969) 470–472.

Z. Y. Wen, L. M. Wu, and Y. Zhang, Set of zeros of harmonic functions of two variables, Harmonic analysis, Tianjin, 1988, Lecture Notes in Mathematics 1494 (Springer, Berlin, 1991) 196–203

will help. I do not know the answer to your specific question.

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 Thank you Andreas for the references. This question it was motivated by the main idea of the Steepest Descent Method and I went curious if I could design a holomorphic function where I could not apply this method. – Leandro Nov 26 2010 at 6:37