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Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
19
votes
Does the proof of Picard's theorem become simpler by increasing the number of points that ar...
Suppose you knew that if $f$ is entire and nonconstant, then $f$ attains all but finitely many values. Then I claim Picard's theorem follows by a completely elementary argument. To show this, suppos …
12
votes
Is this a rational function?
Write $f_0(z)=\sum\frac{z^n}{2^n-1}$ and inductively define $f_k(z)=f_{k-1}(z)-\frac{1}{1-z/2^k}$. By induction, it is easy to see that $f_k$ is given by the power series $$f_k(z)=\sum\frac{z^n}{2^{k …