The Riemann zeta function has one analytic continuation to $\Re(s)>0$ given by: $$\zeta(s)=\frac{s}{s-1}-s\int_{1}^{\infty}\frac{\{x\}}{x^{s+1}}dx \space \space (1)$$ It also has anther analytic continuation to the entire complex plane given by the function that satisfies the functional equation below: $$\zeta(s)=2^{s}\pi^{s-1}\sin(\frac{s\pi}{2})\Gamma(1-s)\zeta(1-s)\space \space (2)$$ and there can be more analytic continuations derived for the zeta function.
So the question is how do I guarantee the uniqueness of the analytic continuation of the zeta function to the domain $\Re(s)>0$ ? Will the identity theorem be sufficient in this case or does the identity theorem will guarantee uniqueness only if analytic continuations are defined in the same domain so identity theorem may not be used to show that analytic continuation in (1) and (2) agree for $\Re(s)>0$