Meromorphic function

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.