Isolated singularity

An isolated singularity at $a$ is a point at which a function is not holomorphic although it is holomorphic throughout some punctured neighbourhood of $a$.

An isolated singularity is removable, a pole, or essential according as its Laurent principal part has zero, finitely many nonzero, or infinitely many nonzero terms.

An isolated singularity is removable when the function extends holomorphically across it, equivalently when its Laurent series has no negative-power terms.

A holomorphic function bounded on a punctured neighbourhood extends holomorphically across the puncture.

A holomorphic function tending to a finite limit at infinity becomes holomorphic at zero after reciprocal substitution.

If $f$ is holomorphic on $0<|z|<R$ andthen the singularity at zero is removable. The Laurent coefficient formula and Cauchy--Schwarz give $|a_{-k}|=O(r^k)$ for each $k\ge 1$, so every principal-part coefficient vanishes as $r\downarrow 0$.

A function has a pole of order $k$ at $a$ when $(z-a{)}^{k}f(z)$ extends holomorphically and is nonzero at $a$.

A simple pole is a pole of order one. Its residue is the nonzero value of the holomorphic extension of $(z-a)f(z)$ at the pole.

A double pole is a pole of order two. If $f(z)=g(z)/(z-a{)}^2$ with $g$ holomorphic near $a$, then ${\mathrm{Res}}_{z=a}f=g^{\prime }(a)$.

A meromorphic function is holomorphic except at isolated poles. Equivalently, it is locally a quotient of two holomorphic functions whose denominator is not identically zero.

A meromorphic function on a Riemann surface is equivalently a holomorphic map to the Riemann sphere, with every pole mapped to infinity. Composition with a rational self-map of the sphere is again such a holomorphic map.

A nonconstant meromorphic function on a compact Riemann surface with exactly one simple pole gives a degree-one holomorphic map to the sphere. A degree-one map is a biholomorphism. The surface therefore has genus zero and the function is a global rational coordinate.

A rational map of the Riemann sphere is a quotient of two complex polynomials, extended to infinity. A nonconstant map has a well-defined positive degree and is a branched covering of the sphere.

Cancel the common polynomial factors of $p$ and $q$ before computing the degree of a rational map of the Riemann sphere. Removable common zeros do not contribute sheets. The reduced maximum polynomial degree is the number of inverse images of a generic target value, counted with positive holomorphic local degrees.

Rotations of the domain and target Riemann spheres act as Möbius transformations represented by SU(2) matrices. A combined symmetry requires the equivariance identity $R(gz)=hR(z)$ for the corresponding transformations. Symmetry of the Wronskian of a rational map alone is insufficient: it tests the ramification directions, not the entire map. Target rotations preserve the angular Jacobian of a rational map, so an equivariant map has a domain-invariant angular density.

For positive integer $n$, $R(e^{i\alpha }z)=e^{in\alpha }R(z)$ and $R(1/z)=1/R(z)$. For $n>1$ its angular Jacobian of a rational map vanishes at the poles and has axial symmetry. Its combined rotational group is the group of rotations preserving an axis; for $n=1$ it enlarges to all domain rotations paired with identical target rotations.

For a reduced rational map $R=p/q$, its derivative is $W/q^2$ with $W=p^{\prime }q-p{q}^{\prime }$. Away from target-coordinate poles, the zeros of this Wronskian identify its ramification points of a holomorphic map; at a pole use the reciprocal coordinate $q/p$. A pole of order $m$ contributes ramification order $m-1$, and infinity must also be checked in local coordinates. For degree $N$ the Riemann-Hurwitz formula gives total ramification $2N-2$. The affine polynomial $W$ can have fewer finite zeros because some ramification lies at infinity. In the rational map approximation for Skyrmions, these directions have zero angular baryon density.

For a nonconstant rational map between round unit Riemann spheres, $R^{\ast }d\Omega =J_{R}d\Omega$. Counting inverse images with multiplicity gives $\int J_{R}d\Omega =4\pi \mathrm{deg}R$. The Jacobian determinant vanishes at ramification points of a holomorphic map, but the integral still records the global degree of a holomorphic map.

A rational function is a quotient $p/q$ of polynomials, with $q$ nonzero. After cancelling common factors, it defines a holomorphic map from the Riemann sphere to itself.

Over a field in which the denominator splits, a proper rational function is a sum of terms $c/(x-a{)}^k$. The coefficient at a simple pole $a$ is obtained by multiplying by $x-a$ and evaluating at $a$.

Let $P$ be a monic polynomial of positive degree $m\ge 1$ with distinct roots of a polynomial $r_j$. The partial fraction decompositionfollows by matching the simple pole residues; the difference is entire and vanishes at infinity. Comparing its expansion there gives ${\sum }_j{r}_j^{\ell }/P^{\prime }(r_j)=0$ for $0\le \ell <m-1$ and ${\sum }_j{r}_j^{m-1}/P^{\prime }(r_j)=1$. These identities give the continuity and derivative jumps needed for constant-coefficient Green functions.

The order of vanishing of a nonzero rational function $f$ at a point $p$ is the exponent of a local parameter in its local factorization. A negative order is the order of a pole.

If $f$ has a pole at $a$, then $e^f$ has an essential singularity there. The exponential series produces infinitely many negative Laurent powers from the nonzero principal part of $f$.

An isolated singularity is essential when its Laurent series has infinitely many nonzero negative-power terms.

Near an isolated essential singularity, a holomorphic function has dense image in the complex plane. If it avoided a disk about $w$, then $1/(f-w)$ would be bounded and extend across the singularity by the Riemann removable singularity theorem. If its extended value is nonzero, $f$ extends holomorphically; if zero, its finite-order zero makes $f$ a pole. Either contradicts an essential singularity.

A singularity is non-isolated when every punctured neighbourhood contains another singularity. An accumulation point of poles or essential singularities cannot itself be classified as a removable singularity, pole, or isolated essential singularity.

The function $1/\cos (1/z)$ has simple poles at $z=1/[\pi (n+1/2)]$, accumulating at zero. Its origin is therefore a non-isolated singularity, despite the underlying cosine of $1/z$ having an isolated essential singularity.