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Let $f,g:\mathbb C \to \mathbb C$ be two analytic functions such that $f(z)(g(z)+z^2)=0$ for all $z$ .Then prove that either $f(z)=0$ or $g(z)=-z^2$.

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    The $z^2$ is only a distraction. $h(z)=g(z)+z^2$ is analytic if $g$ is. (I assume that you forgot to mention that $f$ and $g$ are analytic.) What do you know about the zeros of analytic functions?2011-02-04
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    yes ,f and g both are analytic functions .2011-02-04
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    Please tell me whether i am correct or not:2011-02-04
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    Is Uniqueness theorem applicable to f and this g2011-02-04
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    @N Jana: Yes, that's the result you want to apply. Do you see how to apply it?2011-02-04
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    Suppose that f(a) not zero for some a(C.Then ,by the continuity of f,there exists a disk D(a;d) such that f(z)not 0 in D(a;d).But then h(z)=0 for all z(D(a;d).By the uniqueness theorem ,f(z)=0 in C.2011-02-04
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    @N Jana: Good. (You meant $h$ the last time you wrote $f$.) If you want, now that you have answered your own question, you could post it as an official answer.2011-02-04

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