Are these the Big and Little Picard Theorems

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A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 9.3, 9.4

These seem to be the Big and Little Picard Theorems or at least related to them.

(Exer 9.3) Prove $f$ has an essential singularity at $z_0 \implies \frac 1 f$ has an essential singularity at $z_0$.

(Exer 9.4) Prove that any complex number is arbitrarily close to the image of a nonconstant entire function $f$. Hint: if $f$ is not a polynomial, use Casorati-Weierstrass Theorem (Thm 9.7) on $f(\frac 1 z)$.

Question 1. What exactly are the connections between Exer 9.3 and 9.4 and Big and Little Picard Theorems?

  • In the textbook, 'Picard's Theorem' is only briefly mentioned on p.131. It seems Exer 9.4 is for Little Picard while Exer 9.3 is for Big Picard.

Question 2. Is the following right?

By the contrapositive of Exer 9.4, if $f$ is bounded and entire, then $f$ is either not entire or not nonconstant and thus constant. Thus, both Exer 9.4 and Little Picard are indeed strengthenings of Liouville's Theorem (Thm 5.13).

Possibly related questions: $f$ has an essential singularity in $z_0$. What about $1/f$?, Essential singularlites of the function $f(z)$ and $1/f(z)$?, Proving image of nonconstant, entire function is dense in $\mathbb{C}$

Best Answer

For your first question:

  • These statements are often used in comple analysis textbooks to provide the readers with an intuition of Picard theorems. They are significantly weaker than the theorems, but with the same spirit.

  • Exer 9.3 has a direct corollary that the image of an analytic function $f$ on any punctured neighborhood of its essential singularity $z_0$ is dense in $\mathbb C$. In fact, assume on the contrary that on some punctured neighborhood of $z_0$, $f(z)$ takes value outside the disk $B_r(w)$. Note that $z_0$ is an essential singularity of $f(z)-w$, thus by Exer 9.3 it is a essential singularity of $1/(f(z)-w)$. From our assumption, $|1/(f(z)-w)|\le1/r$, contradicting the fact that an analytic function is unbounded in any punctured neighborhood of its essential singularity. Furthermore, it's immediate from density that $f(z)$ takes value in any disk $B_r(w)$ infinitely often.
    The above statement is a weaker version of

    Picard's great theorem. If an analytic function $f$ has an essential singularity at a point $z_0$, then on any punctured neighborhood of $z_0$, $f(z)$ takes on all possible complex values, with at most a single exception, infinitely often.

  • Exer 9.4 is obviously a weaker version of

    Picard's little theorem. If a function $f:\mathbb C\to\mathbb C$ is entire and non-constant, then there are at most one point which is not in the image of $f$.

For your second question:

  • You are right. Liouville's theorem only claims that the image of an entire non-constant function is unbounded, while Exer 9.4 says it's dense in $\mathbb C$ and Picard's theorem says its complement has at most one point. Both are much stronger than Liouville's theorem. There is, however, some subtleness in saying so:
  • A famous proof of Picard's little theorem (using the modular lambda function) actually needs Liouville's theorem to conclude.
  • Picard's little theorem holds only for entire functions, while Liouville's theorem holds for harmonic functions (which may not even be in two real variables).
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