An isolated singularity of a holomorphic function is removable, a pole, or essential. Three criteria decide the type: the behavior of \(f(z)\) as \(z\to z_0\), boundedness near \(z_0\), and the negative part of the Laurent series.
Isolated and non-isolated singularities
The classification into removable singularities, poles, and essential singularities applies to isolated singularities.
(Branch points)
\(f(z) \coloneqq z^{1\over 2}\) has no holomorphic branch on any punctured disc about \(0\), so \(f\) has no Laurent expansion there and \(z=0\) is not an isolated singularity of a single-valued holomorphic function; it is a branch point. The same holds at \(z = 0\) and \(z = 1\) for \(\qty{z(z-1)}^{1\over 2}\), and at \(z=0\) for \(\Log(z)\).
(Isolated and non-isolated singularities)
A rational function has only isolated singularities, since a nonzero polynomial has finitely many zeros.
\(\Log(z)\) has a singularity at \(z=0\) that is not isolated: every neighborhood of \(0\) meets the branch cut \((-\infty, 0)\), where \(\Log\) is not continuous.
\(G(z) \coloneqq 1/\sin(\pi/z)\) has isolated singularities at the points \(1/n\), \(n\in\ZZ\sm\ts{0}\), and a non-isolated singularity at \(0\), where they accumulate.
The limit criterion
For an isolated singularity \(z_0\) of \(f\):
Removable. \(\lim_{z\to z_0} f(z)\) exists in \(\CC\).
Pole. \(\lim_{z\to z_0} \abs{f(z)} = \infty\).
Essential. Neither limit exists.
The cases are exhaustive and mutually exclusive. For example, \(\sin(z)/z \to 1\) as \(z\to 0\), and \(\abs{1/(z-1)^3}\to\infty\) as \(z\to 1\).
Two paths to \(z_0\) along which \(f\) has different limiting behavior show that neither limit exists. For \(e^{1/z}\) at \(0\): along \(\RR_{>0}\), \(e^{1/z}\to\infty\), and along \(\RR_{<0}\), \(e^{1/z}\to 0\).
The boundedness criterion
By Riemann’s removable singularity theorem, if \(f\) is bounded on a punctured neighborhood of \(z_0\), then the singularity is removable and \(f\) extends holomorphically across \(z_0\). The criterion uses only a bound on \(\abs f\), so it applies when \(\abs f\) can be estimated but not evaluated, for instance to show that a function extends to an entire function.
The Laurent series criterion
Let \(f(z) = \sum_{k\in \ZZ} c_k (z-z_0)^k\) on a punctured disc about \(z_0\).
Removable. \(c_{k} = 0\) for all \(k \leq -1\).
Pole of order \(N\). \(c_{-N}\neq 0\) and \(c_k = 0\) for all \(k<-N\).
Essential. \(c_k \neq 0\) for infinitely many \(k<0\).
This criterion also gives the order of a pole and the residue \(c_{-1}\).
(Order as a valuation)
For \(f(z) = \sum_{k\in\ZZ} a_k(z-a)^k\) about \(a\), let \(v_a(f) \coloneqq \min\ts{k \st a_k\neq 0}\), with \(v_a(f) = -\infty\) when infinitely many negative coefficients are nonzero. Then a zero of order \(n\) has \(v_a(f) = n\), a pole of order \(n\) has \(v_a(f) = -n\), a removable singularity has \(v_a(f) \geq 0\), and an essential singularity has \(v_a(f) = -\infty\).
Summary
| Available information | Criterion |
|---|---|
| an explicit elementary \(f\) | the limit criterion |
| a bound for \(\abs f\) near \(z_0\) | the boundedness criterion |
| a Laurent series, or the order is needed | the Laurent series criterion |
| \(p/q\) with \(p(z_0) = q(z_0) = 0\) and \(q'(z_0)\neq 0\) | the limit criterion: removable, with value \(p'(z_0)/q'(z_0)\) |
| an essential singularity | the Laurent series criterion, or two paths with different limiting behavior |
Singularities at infinity
The type of the singularity of \(f\) at \(z=\infty\) is the type of the singularity of \(g(w) \coloneqq f(1/w)\) at \(w = 0\), and the limit, boundedness, and Laurent series criteria apply to \(g\).