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        riddles >> putnam exam (pure math) >> Lacunary values
        
(Message started by: Icarus on Dec 14^th, 2006, 6:32am)

Title: Lacunary values
Post by Icarus on Dec 14^th, 2006, 6:32am

This one is fairly simple, but I thought I would post it anyway.

A Lacunary value of a complex function is simply a complex number that the function never takes on as a value.

Picard's Great Theorem says that an analytic function with an isolated essential singularity can have at most 1 lacunary value (provided its domain includes an entire deleted neighborhood of the singularity).

Prove the following:
(1) If a meromorphic function has an isolated essential singularity, then it can have at most 2 lacunary values.
(2) Picard's Little Theorem: An entire function has at most 1 lacunary value.
(3) A function which is meromorphic on the entire plane can have at most 2 lacunary values.
(4) None of the results can be bettered (i.e. there exist meromorphic functions with essential singularities and 2 lacunary values).

[edited after a little reading made me realize I had it turned around - the problem used to have the little theorem as given, and asked for a proof of the great theorem. The great theorem is the more fundamental. (Also added the "great" vs "little" terminology, which I had not heard before.)]

Also, does anyone have a good proof for the great theorem?

My own reference (Ahlfors) reproduces Picard's own proof of the little theorem, based on the modular function and the monodromy theorem, but no mention is made of the great theorem. I recall that when I studied C.A. in college, the professor took us instead through a proof of the great theorem, but I now remember no details of it. I've searched through all the links on Google, but the best I was able to find was an assignment to prove it from a result by Montel: A sequence of holomorphic functions all having 0 and 1 as lacunary values must have a subsequence that converges uniformly on compact sets to a function in C^*. The assignment suggested considering the sequence {f(z/n)}, where f has an essential singularity at 0. I'm sure I could work that out, but I am really interested in how the result from Montel is proved, since the deep mathematics is there.