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I'm looking for information on sets of the form $|f(z)| = c$, possibly with additional restrictions (e.g. $|z| < d$). I can probably also assume that $f$ is as nice as you'd like.

I'm interested in things like:

Into how many regions does the curve split the plane?

When is $|f(z)| = c$ path connected?

When is $|f(z)| \leq c$ bounded?

When is $|f(z)| \leq c$ simply connected?

If $(f_n)$ is a sequence of functions, what is the limiting behavior of $|f_n(z)| = c$?

If you could point me toward any references I'd be very grateful. I'd also be interested in references where specific cases are treated, just so I can get an idea of the common techniques.

I know this question is pretty broad but I'm not looking for a particular answer. Any information you could provide would be a great help.

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An interesting case: $|\sin z| = 1/2$. –  GEdgar Aug 30 '11 at 23:58

2 Answers 2

up vote 7 down vote accepted

Assume $c > 0$, and $f$ a nonconstant entire function. $\{z: |f(z)| \le c\}$ is bounded if and only if $f$ is a polynomial. Any bounded component of $\{z: |f(z)| \le c\}$ must contain a zero of $f$. Conversely, if $z_0$ is a zero of $f$, then for $c$ sufficiently small there is a bounded component of $\{z: |f(z)| \le c\}$ in a neighbourhood of $z_0$. On the other hand, $\{z: |f(z)| > c\}$ can't have any bounded components, so any component of $\{z: |f(z)| \le c\}$ is simply connected.

$\{z: |f(z)|=c\}$ is locally a smooth curve except at points where $f' = 0$. If $f'(p) = 0$ and $|f(p)| = c$, then at $p$ there are $k$ smooth curves intersecting, with angles of $\pi/k$ from one to the next, where $f^{(j)}(p) = 0$ for $1 \le j < k$ and $f^{(k)}(p) \ne 0$.

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Thanks Robert, this is great. Is there a reference I should see for properties like these? –  Antonio Vargas Aug 31 '11 at 2:14
    
@Robert: What is a bounded component? –  Weltschmerz Aug 31 '11 at 3:37
    
@Weltschmerz: a connected component that is bounded. –  Robert Israel Aug 31 '11 at 18:17

If $f$ is a polynomial of degree $n$, the number of components of $\{z: |f(z)| \le c\}$ or $\{z: |f(z)| = c\}$ is $n$ minus the number of zeros $p$ of $f'(z)$ such that $|f(p)| \le c$, counted by multiplicity. For example, consider $f(z) = z^4 + z^3 + 1$. $f'(z) = 4 z^3 + 3 z^2$ has a zero of multiplicity 2 at 0, where $|f(0)|=1$, and a zero of multiplicity 1 at $-3/4$, where $|f(-3/4)| = 229/256$. So there are 4 components if $0 \le c < 229/256$, 3 if $229/256 \le c < 1$, and 1 if $c \ge 1$.

Here's an animation of the curve $\{z: |f(z)| = c\}$ in this example, as $c$ varies from $1/2$ to $3/2$. Notice two components merging at $c = 229/256$ and three merging at $c = 1$.

enter image description here

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