prove that $\;f(z) = g(z)\;$ for all $z\in \mathbb{C}$

Question

The problem is:

Let $f(z)$ and $g(z)$ be entire such that $r>0 ,\; f(z) = g(z)$ for all $|z| < r. $

• Prove that $f(z) = g(z)$ for all $z \in \mathbb{C}$

Does that mean I should prove that the function is polynomial ?

I am thinking that's because it's bounded by a polynomial ??

Since $f$ is entire, it is equal to a power series centered at zero with radius of convergence $\infty$, which must match its Taylor series there.

Answer 1

If you consider the function $h(z)=f(z)-g(z)$, then $h(z)$ will also be analytic on the entire $\mathbb{C}$. By the hypothesis, $h(z)=0$ in a disc $|z|<r$. Since the zeroes of an analytic function which does not vanish identically are $\textit{isolated}$, and the disc $|z|<r$ has accumulation points in $\mathbb{C}$ hence the function $h(z)$ is identically $0$ in $\mathbb{C}$.