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Given two entire function f and g, suppose exp(f) , exp(g) and 1 are linearly dependent in the complex field, how can we show that f, g, and 1 are also linearly dependent ?

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  • It might help to start from the definitions of of entire function and linear dependence. – Math1000 Oct 31 '19 at 03:26
  • @Math1000 How does the definition work? I tried to use wronskian, but seemed useless. – Display name Oct 31 '19 at 03:47
  • Well, what does it mean for a set of functions to be linearly dependent? – Math1000 Oct 31 '19 at 03:50
  • @Math1000 Actually, I know the definition of those two terms, but I can’t figure out how it works in the problem. I am not sure how to use the exp to generate f and g. – Display name Oct 31 '19 at 04:19
  • Do you know Picard's (little) theorem? – Daniel Fischer Oct 31 '19 at 12:36
  • @DanielFischer Yeah... It’s about the value of non constant entire function. So, what should I do next ? – Display name Oct 31 '19 at 12:50
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    You can reduce it to this or a simpler situation. – Daniel Fischer Oct 31 '19 at 12:51
  • @DanielFischer Thanks! I think I can deal with it now. – Display name Oct 31 '19 at 13:03
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    @DanielFischer Hi! It’s me again. Through your hint, I have already solved the posted problem. However, I wonder if it is true for arbitrary many entire functions. I tried the same trick by using Picard Thm, but it is not easy as the case of n=2 in this post. Any suggestions will be appreciated. Thanks again. – Display name Oct 31 '19 at 15:25
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    Hmm, for an arbitrary number of functions you cannot conclude that each of the functions is constant. Perhaps from $\sum_{k = 1}^n \exp f_k(z) \equiv 1$ one can conclude that at least one of the $f_k$ is constant, and that of course gives the linear dependence of $1, f_1, \ldots, f_n$. But I don't see how one could prove (or refute) that now. – Daniel Fischer Oct 31 '19 at 16:13

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