Uniqueness of analytic function
Solution 1:
Yes: the Identity Theorem, which states that if $G$ is an open connected subset of $\Bbb C$, if $C\subset G$ has at least an accumulation point in $G$, and if $f,g\colon G\longrightarrow\Bbb C$ are analytic functions such that $f|_C=g|_C$, then $f=g$.