Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
An interesting case: $|\sin z| = 1/2$. – GEdgar Aug 30 '11 at 23:58
up vote 8 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 $f^{(k)}(p) \ne 0$.

share|cite|improve this answer
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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.