For an entire function f such that for each complex number $c$ there is a positive integer $n$ with $f ^{(n)} (c) = 0$. Show that f is a polynomial.

Asked May 03 '22 at 18:16

Active May 03 '22 at 18:16

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If $f$ is an entire function such that for each complex number $c$ there is a positive integer $n$ with $f^{(n)} (c) = 0$. Show that f is a polynomial.

I tried using Taylor series expansion with respect to the point $c$ but that didn't help.

Any hint will also work for me.

asked May 03 '22 at 18:16

Gigabyte

3

This has been asked here not long ago so try a search, but the idea is to use Baire theorem as the sets $F_n$ for which the above holds for precisely $n$ are closed and their union covers the plane, so one must have non empty interior and then identity theorem does the rest – Conrad May 03 '22 at 18:19
@Conrad: There is an even simpler (IMO) proof which does not need Baire's theorem. – Martin R May 03 '22 at 20:14

For an entire function f such that for each complex number $c$ there is a positive integer $n$ with $f ^{(n)} (c) = 0$. Show that f is a polynomial.

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