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While preparing for an exam I got stuck on the following question.

Let $f$ and $g$ be entire functions such that $f(0) = g(0)$ and let $p$ and $q$ be polynomials such that the following equality holds: $e^{f(z)}+p(z) = e^{g(z)}+q(z)$. Prove, using the Picard's Little Theorem, that $f = g$ and $p=q$.

I know the objective is to construct a function which omits two values, yet I don't have any solid idea how to begin. Any hint would be really appreciated.

letreetlneant
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1 Answers1

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Hint: Write the equation as $$ e^{f(z)} - e^{g(z)} = r(z)$$ where $r(z) = q(z) - p(z)$ is a polynomial. Whenever $f(z) - g(z)$ is a multiple of $2 \pi i$, $r(z) = 0$. Now how many such $z$'s will there be?

Robert Israel
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