Infinitesimal calculus MOC

Singular point

In analysis, a singular point or singularity is an input for which a function is not defined. For example, log⁡𝑧 has a singularity at 𝑧 =0 and 𝑧 ↦1𝑧−3𝑖 has a singularity at 𝑧 =3𝑖.¹

Classification of singularities

Removable singularity

A removable singularity is a singularity which may be removed using some kind of holomorphic extension of 𝑓, often by taking the two-sided limit at singular points. For example 𝑓(𝑥) =𝑥2/𝑥 is undefined for 𝑓(0), but it may be holomorphically extended to 𝑓(𝑥) =𝑥 so that 𝑓(0) =0. This is the only possible holomorphic extension, and the value evaluated at the singularity corresponds to the limit lim𝑥→0𝑥2/𝑥.

Poles

A pole is simply a zero (analysis) of a meromorphic function’s reciprocal 1/𝑓(𝑥). For a singularity at 𝑓(𝑧0), we say 𝑧0 is a pole of order 𝑚 if and only if multiplying the function by (𝑧 −𝑧0)𝑚 makes the singularity removable. A pole of order 1 is called a simple pole.

For example, the function 𝑓(𝑧) =1𝑧−𝑧0 has a simple pole at 𝑧0 since lim𝑧→𝑥(𝑧 −𝑧0)𝑓(𝑧) =1 for all 𝑥.

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 𝑧0 Let 𝑓(𝑧) =𝑔(𝑧)ℎ(𝑧) where 𝑔(𝑧) and ℎ(𝑧) are analytic in a neighbourhood of 𝑧0. Let 𝑛 be the smallest non-negative integer such that 𝑔(𝑛)(𝑧0) ≠0, and 𝑚 be the smallest non-negative integer such that ℎ(𝑚)(𝑧0) ≠0. That is to say, 𝑛 is the order of the zero in the numerator and 𝑚 is the order of the zero in the denominator. Then,

order of pole at 𝑧0=𝑚−𝑛

Essential singularity

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

𝑓(𝑧)=𝑒−1/𝑧2

has an essential singularity at 𝑧 =0.

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Footnotes

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