I've recently stumbled upon the following problem:
Let $f:S\to \Bbb{C}$ be analytic on $S:=\{ z: \left| z \right| <10\}$. Let $\Gamma$ be the circle defined as $\Gamma:=\{ z: \left| z\right|=2 \}$. Suppose that $$f(z)=\frac{\sin(z)+z^7-6z}{\exp\{\cos(z)-z+6\}}$$ for all $z\in\Gamma$.
Prove that, for all $z\in S$, $$f(z)=\frac{\sin(z)+z^7-6z}{\exp\{\cos(z)-z+6\}}.$$
So does this question ask to show that $f(z)$, which is valid on $\Gamma$, also holds on $S$? To be honest, I do not understand the nature of this question. If we have a function $f$ with domain $S$, then why does it need to be proved that $f$ is valid on $S$? Or am I misunderstanding the question? Does this question make sense at all?
This question also has the hint which says that one has to show that these two functions have the same Taylor expanstion around $0$. But how come these functoins are "two" if it's the same one single function?
Hint: It's weird to say a function is "valid". Let's try to clear things up: For $z\in \mathbb C,$ define
$$g(z) = \frac{\sin(z)+z^7-6z}{\exp\{\cos(z)-z+6\}}.$$
Then $g$ is an entire function. You are given that $f$ is analytic on $S$ and that $f= g$ on $\{|z|=2\}.$ This situation should look familiar to you.