A meromorphic function is a function that is analytic almost everywhere on the complex plane, except for a set of poles. A meromorphic function can be expressed as the ratio between two entire functions: the poles then occur at the zeroes of the function in the denominator.
Examples of meromorphic functions are all rational functions such as f(z) = (z3-2z + 1)/(z5+3z-1), the functions f(z) = exp(z)/z and f(z) = sin(z)/(z-1)2 as well as the Gamma function and the Riemann zeta function. The functions f(z) = ln(z) and f(z) = exp(1/z) are not meromorphic.