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Let ## f: \mathbb C \rightarrow \mathbb C ## be entire, i.e., analytic in the whole Complex plane. By one of Picard's theorems, ##f ## must be onto , except possibly for one value, called the lacunary value.

Question: say ##0## is the lacunary value of ##f ##. Must ## f ## be of the form ##e^{g(z)}## , with ##g(z)## analytic?.

Clearly if ##g(z)## has no lacunary values, then ## e^{g(z)}## will have only ##0## as its lacunary value,

and if ##g(z) ## has only ##w## as its lacunary value, then this value will be assumed in ## w+i2\pi n ##, so ##0## will still be the lacunary value of ## e^{g(z)}##. Maybe we can consider composing functions with known lacunary values, but I don't see offhand how, since I don't know the lacunary values of general entire functions.

I considered using Weirstrass factorization thm, but it seems overkill and has led nowhere.

Thanks.

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# Entire Functions and Lacunary Values.

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