# For a power series function that is constantly zero prove that the coefficients are zero

Suppose that power series function $a_0 + a_1x + a_2x^2 + a_3x^3 + ...$ is constantly zero on a bounded non-empty open interval $I$, which may or may not contain $0$.

Prove that $a_j = 0$ for every $j$, so that the power series is constantly zero on its whole domain (which is of course centered at $0$).

Here is my thought: let us call power series function given in the problem $f$. Let $I = (a, b)$ (where $a < b$, of course), and let $c= (a+b)/2=$ the center of $I$. I don't know how to go from here. Any help?

• I've noticed that you have asked 6 questions in two days. I wanted to make sure that you are aware of the quotas 50 questions/30 days and 6 questions/24 hours, so that you can plan posting your questions accordingly. (If you try to post more questions, StackExchange software will not allow you to do so.) For more details see meta. Mar 30 '16 at 16:41

Complex analytic solution:

$f$ is a power series, so it is analytic. Since $f$ is $0$ in a uncountable set, it agrees with $0$ everywhere in its domain.

Real analytic solution:

There exists a unique set of coefficients $(b_j)_{j=0}^\infty$ such that $$f(x) = \sum_{j=0}^\infty b_j (x - c)^j = \sum_{j=0}^\infty \frac{f^{(j)}(c)}{j!} (x - c)^j$$ Since the derivatives of $f$ at $c$ are all $0$, all $b_j = 0$. $f$ is a power series, so $(a_j)_{j=0}^\infty$ is a unique set of coefficients so that $$\sum_{j=0}^\infty a_j x^j = \sum_{j=0}^\infty b_j (x-c)^j$$ and it turns out that $a_j = 0$.

• You have to justify why you can use the identity principle, however. Mar 30 '16 at 0:43
• @PedroTamaroff Thanks. It is fixed. Mar 30 '16 at 0:44
• I havent done any complex analysis. Can you explain the first step a little more?
– user326676
Mar 30 '16 at 0:45
• @THT16224 The first way uses the identity theorem. It is justified since every power series is analytic (a.k.a. holomorphic) Mar 30 '16 at 0:46
• I meant the first step in the real analytic way
– user326676
Mar 30 '16 at 0:52