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Amazing property of entire functions


Prove that if p,q,r are non-constant, non-vanishing entire functions with p+q+r=0 then there exists an entire function h such that p,q,r are constant multiples of h.

Solution
Because each of the given function do not vanish, we can write \frac{p}{r}+\frac{q}{r}=-1. Now look at the function \frac{p}{r}. This is an entire function, as a quotient of two non-vanishing holomorphic functions. Because \frac{q}{r} \neq 0 also, we find that \frac{p}{r} avoids two complex numbers, namely 0 and -1. A weak variant of Picard’s Theorem says that if this happens, then \frac{p}{r} is constant, and so is \frac{q}{r}. Therefore, p,q,r are scalar multiples of r.

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