# Thread: prove a certain function is a polynomial

1. ## prove a certain function is a polynomial

I have seen the following problem and I didn't manage to solve it, so I will appreciate any help.

Suppose f is an analytic function defined everywhere in $\mathbb{C}$ and such that for each $z_0\in\mathbb{C}$ at least one coefficient in the expansion f(z)=\sum_{n=0}^\infty c_n(z-z_0)^n is equal to 0. Prove that f is a polynomial.

It is easy to prove that for each z_0 there exists n such that f^{(n)}(z_0)=0, which means that each z_0 is root for a derivative of f, but how do I conclude that f s a polynomial?

2. ## Re: prove a certain function is a polynomial

If f(*) is analytic everywhere in $\mathbb{C}$ , then its Taylor expansion in any point $z=z_{0}$ is...

$f(z)= \sum_{n=0}^{\infty} a_{n}\ (z-z_{0})^{n}$ (1)

... where...

$a_{n}= \frac{1}{n!}\ \frac{d^{n}}{d z^{n}} f(z)_{z=z_{0}}$ (2)

Now if $\vorall z_{0}$ is $a_{n}=0$ then the derivative of order n of f(*) vanishes everywhere and that means that f(*) is a polynomial of order k<n...

Kind regards

$\chi$ $\sigma$

3. ## Re: prove a certain function is a polynomial

you can solve it using 2 hints:

-whats the union of sets {f^(k)(z)=0} where k runs through natural numbers?
-what are the properties of the zero set of a holomorphic function?

4. ## Re: prove a certain function is a polynomial

I was thinking something like this: the union of sets {f^(k)(z)=0} where k runs through natural numbers is uncountable from the hypothesis, that means that there must be a number N which satisfies f^(N)=0 for all z, which demonstrates that f is a polynomial. But I am not sure if there are some missing arguments.

About the set of the zeros of a holomorphic function, I suppose it must be finite? If a holomorphic function has countable many zeros, that means that it is the function constant=0?

5. ## Re: prove a certain function is a polynomial

Put $F_n:=\{z\in\mathbb C, f^{(n)}(z)=0\}$. $F_n$ is a closed subset of $\mathbb C$ and $\bigcup_{n\in\mathbb N}F_n=\mathbb C$. Using Baire category's theorem, we can find $n_0$ such that $F_{n_0}$ has a nonempty interior. We can find a ball on which $f^{(n_0)}$ is identically $0$ on this ball. Now you can conclude, using the fact that the zeros of an holomorphic function are isolated.