Three types
Definition. Let \(f\) be holomorphic on \(0<|z-a|<r\). The point \(a\) is removable if \(f\) extends holomorphically to \(a\), a pole if \(|f(z)|\to\infty\), and essential otherwise.
Removable singularities
Theorem. A singularity is removable if and only if \(f\) is bounded near it. Equivalently, \((z-a)f(z)\to0\). The extension value is the limit forced by Cauchy's formula.
Poles and principal parts
Proposition. The point \(a\) is a pole of order \(m\) if and only if \(f(z)=(z-a)^{-m}g(z)\) with \(g\) holomorphic and \(g(a)\ne0\). The finite sum of negative powers in the Laurent expansion is the principal part.
Casorati-Weierstrass theorem
Theorem. Near an essential singularity, the image of every punctured neighborhood is dense in \(\mathbb C\). Essential singularities are therefore maximally unstable compared with removable singularities and poles.
Classification by reciprocal
Example. The singularity of \(1/f\) at a zero of order \(m\) of \(f\) is a pole of order \(m\). The function \(e^{1/z}\) has an essential singularity at \(0\), with infinitely many negative Laurent coefficients.
Local algebra
Remark. The classification is algebraic once Laurent series exist: removable means no negative powers, pole means finitely many negative powers, essential means infinitely many negative powers.
The trichotomy
Theorem. Let \(f\) be holomorphic on \(0<|z-a|<R\). Exactly one of the following holds:
- \(f\) is bounded near \(a\) and extends holomorphically;
- \(|f(z)|\to\infty\), and \(a\) is a pole;
- the image of every punctured neighborhood is dense in \(\mathbb C\), and \(a\) is essential.
Proof. Consider the Laurent expansion \(f(z)=\sum_{n\in\mathbb Z}a_n(z-a)^n\). No negative coefficients gives a removable singularity. Finitely many negative terms gives a pole, with the lowest exponent determining its order. Infinitely many gives an essential singularity. The density conclusion is Casorati--Weierstrass, proved below.
Reciprocal characterization of poles
Proposition. \(f\) has a pole of order \(m\) at \(a\) if and only if \(1/f\) extends holomorphically with a zero of order \(m\) at \(a\).
Proof. A pole has \(f(z)=(z-a)^{-m}g(z)\) with \(g(a)\ne0\), so \(1/f=(z-a)^m/g(z)\). The converse follows by inversion.
Casorati--Weierstrass
Theorem. If \(a\) is an essential singularity of \(f\), then \(f(0<|z-a|<r)\) is dense in \(\mathbb C\) for every \(r>0\).
Proof. If some disc \(D(w,\varepsilon)\) were omitted near \(a\), then \[ g(z)=\frac1{f(z)-w} \] would be bounded and hence extend holomorphically to \(a\). If \(g(a)\ne0\), then \(f=w+1/g\) extends, making the singularity removable. If \(g(a)=0\), then \(1/g\) has a pole, making \(f\) have a pole. Both contradict essentiality.
Great Picard theorem
A stronger theorem states that near an essential singularity, \(f\) assumes every complex value, with at most one exception, infinitely often. The example \(e^{1/z}\) omits \(0\) and takes every nonzero value infinitely often. Picard's theorem requires normal-family or covering-space methods beyond Casorati--Weierstrass.
Behavior at infinity
The point \(\infty\) is studied through \(g(w)=f(1/w)\) at \(w=0\). An entire function has a removable singularity at infinity exactly when it is constant, a pole exactly when it is a polynomial, and an essential singularity exactly when it is transcendental entire.