- #1
MathLearner123
- 25
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I want to prove following (Big Picard Theorem forms):\
Theorem.
The followings are equivalent:\
a) If ##f \in H(\mathbb{D}\setminus\{0\})## and ##f(\mathbb{D}') \subset \mathbb{C} \setminus \{0, 1\}##, then ##f## has a pole of an removable singularity at ##0##.\
b) Let ##\Omega \subset \mathbb{C}## is a open subset, ##f : \Omega \to \mathbb{C}## is holomorphic and ##z_0 \in \mathbb{C}##. If ##f## has an essential singularity at ##z_0##, then, with at most one exception, ##f## attains every complex value infinitely many times;\
c) Let ##f : \mathbb{C} \to \mathbb{C}## a entire function which is not polynomial. Then, with at most one exception, ##f## attains every complex value infinitely many times;
I have proved that a) ##\implies## b) ##\implies## c) and that b) ##\implies## a) but I don't know how to start proving that c) implies a) or b). And another thing: Is mathematically correct to say that those points are equivalent? Thanks!!
Theorem.
The followings are equivalent:\
a) If ##f \in H(\mathbb{D}\setminus\{0\})## and ##f(\mathbb{D}') \subset \mathbb{C} \setminus \{0, 1\}##, then ##f## has a pole of an removable singularity at ##0##.\
b) Let ##\Omega \subset \mathbb{C}## is a open subset, ##f : \Omega \to \mathbb{C}## is holomorphic and ##z_0 \in \mathbb{C}##. If ##f## has an essential singularity at ##z_0##, then, with at most one exception, ##f## attains every complex value infinitely many times;\
c) Let ##f : \mathbb{C} \to \mathbb{C}## a entire function which is not polynomial. Then, with at most one exception, ##f## attains every complex value infinitely many times;
I have proved that a) ##\implies## b) ##\implies## c) and that b) ##\implies## a) but I don't know how to start proving that c) implies a) or b). And another thing: Is mathematically correct to say that those points are equivalent? Thanks!!
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