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, Exer 9.4, Little Picard or both is indeed a strengthening 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 $\mathbb{C}$

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  • 1
    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. – Cave Johnson 11 hours ago
  • @CaveJohnson thanks! Is 9.3 for big and 9.4 for little? – BCLC 11 hours ago

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