Uniqueness theorem for Rational Functions

Question

I know that for polynomials $P,Q$, the equation $P(z) \equiv Q(z)$ implies that they are of the same degree and have the same coefficients. Is there an analogous result for rational fucntions? That is, if $R,S$ are two rational functions and $R(z)=S(z)$ for all $z$ what is the relationship between $R$ and $S$?

Thank you

Answer 1

If $R(z) = S(z)$ for all $z$ such that this equation makes sense, then $R(z) - S(z) = 0$ for all $z$. Now, $R-S$ is a rational function that's equal to zero everywhere where it is defined. You can easily show that such function is given by $0/Q(z)$ where $Q$ is some polynomial.