In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior. Complex analysis is the branch of mathematics investigating functions of complex numbers. ... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ...

Formally, consider an open subset U of the complex plane C, an element a of U, and a holomorphic function f defined on U - {a}. The point a is called an essential singularity for f if it is a singularity which is neither a pole nor a removable singularity. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In complex analysis, a removable singularity of a function is a point at which the function is not defined (a singularity) but at which the function can be defined without creating any problems. ...

For example, the function f(z) = exp(1/z) has an essential singularity a = 0. The exponential function is one of the most important functions in mathematics. ...

The point a is an essential singularity if and only if the limit In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes larger and larger; or the behavior of a sequences elements, as their index becomes larger and larger. ...

does not exist as a complex number nor equals infinity. This is the case if and only if the Laurent series of f at the point a has infinitely many negative degree terms (the principal part is an infinite sum). Infinity refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. ... A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ... In mathematics, the principal part of the Laurent series of a function f(z), is the series of terms with negative degree, that is Say f(z) has an essential singularity at p. ...

The behavior of holomorphic functions near essential singularities is described by the Weierstrass-Casorati theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity a, the function f takes on every complex value, except possibly one, infinitely often. The Weierstrass-Casorati theorem in complex analysis describes the remarkable behavior of holomorphic functions near essential singularities. ... In complex analysis, mathematician Charles Émile Picards name is given to two theorems regarding the range of an analytic function. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...