This construction is helpful in the study of holomorphic and meromorphic functions. Mathematically, Jacobian elliptic functions are doubly periodic meromorphic functions on the complex plane. For example, formula 4 is a meromorphic function on the two-dimensional complex affine space. Since the poles of a meromorphic function are isolated, there are at most countably many. The Riemann zeta function is meromorphic with a single pole of order one at . Thereby the notion of a meromorphic function can be defined for every Riemann surface. In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles. We can show functional equations and meromorphic continuations of these functions. Herman rings for transcendental meromorphic functions have been studied by T. Nayak. A meromorphic function may have infinitely many zeros and poles. Rational functions are representative examples of meromorphic functions. The Laplace transform of the Heaviside step function is a meromorphic function. For meromorphic functions, particularly, there is a duality between zeros and "poles". Instead, modular functions are meromorphic. This is an equality of meromorphic functions valid on the whole complex plane.