Picard's theorem says that if $f\colon D\to \mathbb{C}$ is an entire nonconstant function or holomorphic with an essential isolated singularity, then $\mathbb{C}\setminus f({D})$ is either empty or a singleton. In the meromorphic case this set of exceptional values is of cardinality at most two (including $\infty$).

Q: What can we say about $\mathbb{C}\setminus f({D})$ if $f\colon D\to \mathbb{C}$, $D\subset \mathbb{C}$ is nonempty open and connected, is holomorphic:

  1. If $n\ge 3$, then does there exist $f$ such that $\mathbb{C}\setminus f({D})$ is of cardinality $n$?

  2. Does there exist $f$ such that $\mathbb{C}\setminus f({D})$ is of infinite but countable cardinality?

  3. Does there exist $f$ such that $\mathbb{C}\setminus f({D})$ is a meagre set or of Lebesgue measure zero but of uncountable cardinality?

  • $\begingroup$ What is a singularity which is not isolated? $\endgroup$
    – abx
    Jan 28, 2014 at 11:02
  • $\begingroup$ Branch like $log$ or $x^{1/3}$. But I'll clarify by omitting that. $\endgroup$ Jan 28, 2014 at 11:03

1 Answer 1


It is clear that $f(D)$ is connected.

If $D$ is the unit disc, then every closed set whose complement is connected can be exceptional. Let $F$ be a closed set with connected complement containing at least 2 points. If $C\backslash D$ is not simply commented, then the universal covering map $f:D\to C\backslash F$ takes every value infinitely many times, and does not take values in $F$. If $C\backslash F$ is simply connected, construction is easier. If $F$ is one point, again we have easy examples.

Same happens when $D$ is any simply connected set, other that $C$, by the Riemann mapping theorem. So the answer to all your questions is "yes" if $D$ is simply connected but different from $C$.

If $E=C\backslash D$ is a finite set, then the number of exceptional values at every point of $E$ is at most one, and these exceptional values can be arbitrarily prescribed. So we can obtain any finite set of cardinality at most that of $E$.

If $D$ is a complement of a Cantor set $E$, then the answer is more complicated and not all possibilities were studied. Carleson gave an example of a cantor $E$, or zero capacity such that every analytic function outside $E$, for which $E$ is singular, can omit at most 5 values, and 5 is best possible for this $E$.

  • $\begingroup$ Thanks for that. My supplementary question is in all these cases can $f$ be analytically (not) continued to cover the exceptional values. I have in my work badly behaved functions whose exceptional values are meagre and I'm trying to understand these. $\endgroup$ Jan 30, 2014 at 3:12
  • $\begingroup$ In all these questions, one can have $f$ non-continuable from $D$. If $F$ contains no analytic arcs, this is automatic. If $F$ has analytic boundary arcs one can achieve this by superposition with an appropriate function which is not continuable. $\endgroup$ Jan 30, 2014 at 4:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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