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Plot of the function exp(1/z), centered on the essential singularity at z=0. The hue represents the complex argument, the luminance represents the absolute value. This plot shows how approaching the essential singularity from different directions yields different behaviors (as opposed to a pole, which, approached from any direction, would be uniformly white).
Model illustrating essential singularity of a complex function 6w=exp(1/(6z))
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.
exists but does not exist, then a is a zero of f and a pole of 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.)