The category essential singularity is a "left-over" or default group of isolated singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner – removable singularities and poles.
Consider an open subset of the complex plane . Let be an element of , and a holomorphic function. The point is called an essential singularity of the function if the singularity is neither a pole nor a removable singularity.
For example, the function has an essential singularity at .
- and exist, then a is a removable singularity of both f and 1/f.
- does not exist but exists, then a is a pole of f and a zero of 1/f.
- nor exists, then a is an essential singularity of both f and 1/f.
Another way to characterize an essential singularity is that the Laurent series of f at the point a has infinitely many negative degree terms (i.e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of .
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass 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 many times. (The exception is necessary, as the function exp(1/z) never takes on the value 0.)
- Weisstein, Eric W. "Essential Singularity". MathWorld, Wolfram. Retrieved 11 February 2014.