meromorpic

Meromorphic refers to a type of function in complex analysis. A function is called meromorphic if it is analytic (holomorphic) everywhere in its domain except for a set of isolated points, where it may have poles. These poles are a specific type of singularity where the function's value tends towards infinity.

More formally, a function f is meromorphic on an open set U in the complex plane if for every point z₀ in U, there exists a neighborhood of z₀ where f is analytic except possibly at a finite number of points, and at each of these points, f has a pole. This means that in a neighborhood of a pole z₀, the function can be represented by a Laurent series of the form f(z) = ∑_{n=-k}^∞ a_n (z-z₀)^n, where k is a positive integer and a_{-k} ≠ 0. The integer k is called the order of the pole.

Functions that are meromorphic on the entire complex plane are sometimes called entire meromorphic functions. The set of meromorphic functions on a domain forms a field, meaning that sums, differences, products, and quotients (where the denominator is not identically zero) of meromorphic functions are also meromorphic. This property makes them a fundamental class of functions in complex analysis, appearing in areas such as the theory of elliptic functions and the study of Riemann surfaces.