Infinitesimal calculus MOC

Singular point

In analysis, a singular point or singularity is an input for which a function is not defined. For example, has a singularity at and has a singularity at .¹

Classification of singularities

Removable singularity

A removable singularity is a singularity which may be removed using some kind of holomorphic extension of , often by taking the two-sided limit at singular points. For example is undefined for , but it may be holomorphically extended to so that . This is the only possible holomorphic extension, and the value evaluated at the singularity corresponds to the limit .

Poles

A pole is simply a zero (analysis) of a meromorphic function’s reciprocal . For a singularity at , we say is a pole of order if and only if multiplying the function by makes the singularity removable. A pole of order 1 is called a simple pole.

For example, the function has a simple pole at since for all .

The order of a pole can be determined by reducing each term with the leading order term of its Laurent series, or equivalently

To calculate the order of a pole at Let where and are analytic in a neighbourhood of . Let be the smallest non-negative integer such that , and be the smallest non-negative integer such that . That is to say, is the order of the zero in the numerator and is the order of the zero in the denominator. Then,

Essential singularity

An essential singularity is a singularity that is not a pole (and is not removable).

has an essential singularity at .

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Footnotes

1. 2023. Advanced Mathematical Methods, p. 54 ↩