Uniqueness principle theorem :
If $f$ and $g$ are analytic functions on a domain $D$, and if $f(z)=g(z)$ for $z$ belonging to a set that has a non isolated point, then $f(z)=g(z)$ for all $z\in D$.
Now my question is this:
If $f$ and g are analytic on a domain $D$, and if $f(z)=g(z)$ for infinitely many $z\in D$, Is it always true that $f\equiv g$ on $D$?
I think this is not true. As we know that an analytic function on some disk has a Taylor expansion in that disk. And If we expand any given analytic function (by using Taylor series) in a smaller disk than the given domain. By doing this we will get another function whose domain will be smaller than that of the given function but they will be equal on infinite many values.
This is a vague idea. Please tell me if its right. If not then correct me by giving some appropriate example to support the above statement.
Thanks.
The assertion is false, for example take any countable set like $\pi\mathbb Z$ and functions like $$\sin(x), 2\sin(x)$$ they are unequal, but $\sin(x) = 2\sin(x) = 0 \quad\forall\ x\in \pi\mathbb Z$