Essential singularity

In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior.

The category essential singularity is a "left-over" or default group of 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 U of the complex plane C. Let a be an element of U, and f : U \ {a} → C a holomorphic function. The point a is called an essential singularity of the function f if the singularity is neither a pole nor a removable singularity.

For example, the function f(z) = e^1/z has an essential singularity at z = 0.

Let a be a complex number, assume that f(z) is not defined at a but is analytic in some region U of the complex plane, and that every open neighbourhood of a has non-empty intersection with U.

If both

If

Similarly, if

If neither

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 ${\displaystyle a}$ for which ${\displaystyle f(z)(z-a)^{n}}$ is not differentiable for any integer ${\displaystyle n>0}$, then ${\displaystyle a}$ is an essential singularity of ${\displaystyle f(z)}$.

...
Wikipedia