Meromorfian

Meromorfian is a term used in mathematics, specifically within the field of complex analysis. It describes a property of functions that are defined on a domain within the complex plane. A function is considered meromophian if it can be expressed as the ratio of two holomorphic functions, where the denominator is not identically zero. Holomorphic functions are complex-valued functions that are differentiable in the complex sense. This means that at every point in its domain, the function has a well-defined complex derivative.

The existence of poles is a key characteristic of meromophian functions. While holomorphic functions are smooth and well-behaved everywhere in their domain, meromophian functions are allowed to have isolated singularities known as poles. At a pole, the function's value tends towards infinity. These poles must be "removable" in the sense that they are not essential singularities or branch points. Essentially, meromophian functions are "nice" functions with a controlled set of singularities.

The set of meromophian functions on a given domain forms a field, which means that sums, differences, products, and quotients (where the denominator is not zero) of meromophian functions are also meromophian. This property makes them a fundamental object of study in complex analysis, as they encompass a broad class of functions that appear in various mathematical and physical applications, including the study of differential equations and the theory of functions of several complex variables.