which is also a meromorphic function of "m", we see that it has exactly the same poles and zeroes as the quotient of determinants we are trying to compute: if "m" is an eigenvalue of the operator number one, then will be an eigenfunction thereof, meaning ; and analogously for the denominator. By Liouville's theorem, two meromorphic functions with the same zeros and poles must be proportional to one another. In our case, the proportionality constant turns out to be one, and we get