I was wondering what can be said about two complex functions that are very close on a domain?
For example let, f,g be two analytic function with $|f-g| = O(e^{-X})$ on the real interval $(X, 2X],$ where $X$ is a large number.
Could we say that they are equal or almost equal on a large domain in $\mathbb{C}$ containing the interval? Basically is it true to concluse that $g(z)=f(z)+ ce^{-z}$ on $\Re(z)> X$?
The identity theorem says that if two analytic complex functions are equal on a set with an accumulation point then they are equal everywhere else in the domain. The proof uses their Taylor expansion around the accumulation point.
Edit: After Jose's answer I realized that I need to consider a domian with bound hight, so we are looking at $\Re(z)> X, \Im(z) \ll 1.$
You have $\bigl|\sin(x)e^{-x}-0\bigr|=O(e^{-x})$ on $(x,2x]$. On the other hand, the restriction of $\sin(z)e^{-z}$ to any vertical line is unbounded. Therefore, you certainly don't have $\sin(z)e^{-z}=0+e^{-z}$ on any half-plane $\{z\in\Bbb C\mid\operatorname{Re}z>x\}$.