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TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26
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Every isolated singularity is removable, a pole, or essential

Statement

Let f be holomorphic on a punctured disc 0<∣z−a∣<R. Then exactly one of the following holds:

  1. a is a removable singularity of f;
  2. a is a pole of f;
  3. a is an essential singularity of f.

Equivalently, if

f(z)=∑n∈Zcn(z−a)n,

then the three cases are: no negative coefficients, finitely many negative coefficients but not all zero, or infinitely many negative coefficients.

Facts & Assumptions

Given: A function f holomorphic on 0<∣z−a∣<R and its Laurent expansion there.

[L1]

A removable singularity is exactly the case of zero principal part (Characterizations of removable singularities).

[L2]

A pole is exactly the case of a finite nonzero principal part (Characterizations of poles).

[L3]

An essential singularity is, by definition, an isolated singularity that is neither removable nor a pole (Isolated singularities: removable, poles, and essential singularities).

[L4]

Every holomorphic function on a punctured disc has a Laurent expansion there (Laurent expansion on an annulus).

Proof

technique · direct
1.1

By [L4], the Laurent expansion exists, and its set of negative coefficients is either empty, finite nonempty, or infinite.

givenL4
2.1

If there are no negative coefficients, the principal part is zero, so [L1] makes the singularity removable.

step 1.1L1
2.2

If there are finitely many negative coefficients and at least one is nonzero, the principal part is finite and nonzero, so [L2] makes the singularity a pole.

step 1.1L2
3.1

If there are infinitely many negative coefficients, the singularity is neither removable nor a pole by steps 2.1 and 2.2, so [L3] makes it essential.

step 1.1step 2.1step 2.2L3
4.1

The three coefficient cases are mutually exclusive and exhaustive, and steps 2.1 through 3.1 identify them with the three singularity types.

step 1.1step 2.1step 2.2step 3.1∎

Depends on

Used by

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Sources