Classification of isolated singularities

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.

This criterion characterizes a removable singularity of a holomorphic function; the criterion is a theorem about the singularity type.

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 pole is the negative-order case in the zeros and poles classification of a meromorphic function.

A function holomorphic outside a disk has a pole of positive integer order $N$ at infinity when its reciprocal-coordinate representative $f(1/w)$ has a pole of that order at $w=0$. Equivalently $f(z)=z^{N}g(1/z)$ with $g$ holomorphic and nonzero at zero. The Laurent series about infinity is interpreted in this coordinate: finitely many positive powers of $z$ form its principal part. For example, $z^N{e}^{1/z}=z^N+z^{N-1}+\cdots$ has a pole of order $N$ at infinity despite its essential singularity at zero.

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.

For holomorphic $N,D$ with $N(z_0)D(z_0)\ne 0$, write the quotient as $h(z)/(z-z_0)$ where $h=N/D$. Its Taylor series immediately gives the displayed Laurent series. The residue is $h(z_0)$ and the constant regular term is $h^{\prime }(z_0)$; the latter requires differentiating both numerator and remaining denominator. Higher regular coefficients follow from the higher Taylor coefficients of $h$.

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.

For a meromorphic function viewed as a holomorphic map to the unit Riemann sphere, this is the length density of the pulled-back spherical metric. Its square is the pulled-back area density. At a pole use the coordinate $1/f$, which gives the same density. Some conventions omit the factor two and use a sphere of area $\pi$; all area normalizations must then change accordingly.

A meromorphic function on a punctured disk has finite pulled-back spherical area precisely when it extends to the Riemann sphere at the puncture. A removable singularity or pole gives a smooth spherical extension and finite area. At an essential singularity, the meromorphic version of the Great Picard theorem gives infinitely many preimages of all but at most two spherical targets. The area formula then makes the area infinite. The logarithmically weighted annular area divided by $\log (1/r)$ tends to the total unweighted area divided by $4\pi$, by averaging a nondecreasing area function.

A meromorphic function $f$ has an algebraic addition theorem if some nonzero polynomial $P\in \mathbb{C}[U,V,W]$ satisfies $P(f(z),f(w),f(z+w))=0$ wherever the three values are finite. For a polynomial $f$ of degree $d>0$, take roots $z_i$ of $f(z)-U$ and $w_j$ of $f(w)-V$; the product ${\prod }_{i,j}(W-f(z_i+w_j))$ has polynomial coefficients in $U,V$ by the Fundamental theorem of symmetric polynomials, is monic in $W$, and supplies an addition relation. An elliptic function also has such a relation, by finite algebraic extension of its function field and the Weierstrass elliptic function addition formula.

The displayed entire function has exactly the periods $2\pi i\mathbb{Z}$: a period $c$ would make $e^z(e^c-1)$ a constant integer multiple of $2\pi i$, forcing $e^c=1$. If a polynomial addition relation existed, specialize its middle argument at a point with $e^c=\alpha >0$ irrational, avoiding the finitely many specializations making the entire polynomial vanish. With $t=e^z$, the relation becomes a nonzero finite sum of exponentials $e^{(m+\alpha n)t}$. These real exponents are distinct, so the largest one dominates as $t\to +\infty$, a contradiction. This demonstrates that an algebraic addition theorem is stronger than simple periodicity.

Poles are isolated, including in the coordinate $w=1/z$ around infinity. Compactness of the Riemann sphere therefore leaves only finitely many poles. Subtract the finite principal parts at finite poles and the polynomial principal part at infinity. The remainder is entire with a finite limit at infinity, hence bounded and constant by Liouville's theorem. The original function is a sum of a polynomial and finitely many partial fractions, so is rational.

Mittag-Leffler's theorem prescribes the principal parts of a meromorphic function. Given an open subset $D$ of the complex plane, a subset $A$ having no accumulation point in $D$, and a finite sum of negative powers $P_a(z)={\sum }_{j=1}^{m_a}{c}_{a,j}(z-a{)}^{-j}$ for each $a\in A$, there is a meromorphic function on $D$ with exactly these principal parts and no other poles. The difference of any two such meromorphic functions is a holomorphic function on $D$, because their principal parts cancel at every prescribed pole.

A meromorphic function on the plane with $|f(z)|\le C|z{|}^n$ outside a disc has no poles there and only finitely many poles inside. Subtract their finite principal parts to obtain an entire function. For $n\ge 0$, Cauchy estimates make every derivative of order above $n$ vanish, so the remainder is a polynomial. For $n<0$ the remainder tends to zero at infinity and is zero by Liouville theorem. Adding back the principal parts gives a rational function in either case.

Nevanlinna theory studies the growth and value distribution of meromorphic functions through their integrated pole counts and logarithmic proximity to target values. The Nevanlinna characteristic combines both quantities, while the Nevanlinna first main theorem compares it with the corresponding counts and proximity for any fixed target.

This theory compares the area, boundary length and Euler characteristic of a surface mapped holomorphically to another surface, using the pulled-back metric. It extends covering-degree and Riemann-Hurwitz formula estimates when some boundary maps into the interior of the target. The Ahlfors second fundamental theorem and Ahlfors five islands theorem are important consequences.

For a nonconstant meromorphic function on the plane, let $S(R)$ be its normalized pulled-back spherical area on $|z|<R$, and $L(R)$ the spherical length of its boundary image counted with multiplicity. The Cauchy-Schwarz inequality gives $L(R{)}^2\le 8{\pi }^{2}R{S}^{\prime }(R)$. If $L/S$ stayed above a positive constant for all large $R$, integration of $(1/S{)}^{\prime }\le -c/R$ would make $1/S$ negative. Thus there are arbitrarily large radii with $L/S$ arbitrarily small, even when the total area is finite.

For a finite simply connected covering surface over the Riemann sphere and $q\ge 3$ fixed separated Jordan domains, let $n_j$ count its simply connected islands over the $j$th domain once each, regardless of mapping degree. Let $S$ be its average sheet number and $L$ its spherical boundary length. The displayed inequality holds with $h$ depending only on the target domains. For regular bordered surfaces $f:X\to Y$, a general formulation is $\max \{-\chi (X),0\}\ge -S\chi (Y)-hL$, where $L$ counts only the boundary mapped into the interior of $Y$ and $\chi$ is the modern Euler characteristic. The boundary error disappears for an actual branched covering. The covering-surface formulation is explained in www.math.purdue.edu/~eremenko/dvi/ahl.pdf ; the simply connected island estimate is also recorded in Lemma 2 of pdfs.semanticscholar.org/d247/562607e237af7c8fc6e683b77be6b94829fa.pdf .

Every nonconstant meromorphic function on the complex plane has a simple island over at least one of any five Jordan domains on the Riemann sphere with disjoint closures. The Ahlfors second fundamental theorem, area comparison over target domains, and a length-area exhaustion of the complex plane prove this: without a simple island every counted island has degree at least two, forcing $3S\le (5/2)S+O(L)$.

An island over a Jordan domain $D$ is a relatively compact component $U$ of its inverse image on which the meromorphic function is a proper map onto $D$. A component reaching the boundary of the source is instead a peninsula. The island has an integer mapping degree; degree one means that the map is a biholomorphism. A simply connected island can have larger degree and need not be a simple island.

A simple island is an island of a meromorphic function of mapping degree one. It is mapped biholomorphically onto the target Jordan domain. This is stronger than the source island being simply connected.

For a holomorphic map between metric Riemann surfaces, the average sheet number is the pulled-back source area divided by the target area. The area formula identifies it with the average number of preimages counted with multiplicity. The same definition over a target region $D$ gives $S_D=A_f(f^{-1}D)/A(D)$.

For distinct fixed target values and a nonconstant meromorphic function, the truncated second main theorem has the displayed form, with $S_f=O({\log }^{+}{T}_f+\log r)$ outside a set of finite linear measure. The truncated Nevanlinna counting function retains only one unit at each preimage. For finite-order functions the error is $O(\log r)$; for transcendental functions it is $o(T_f)$ along the nonexceptional radii.

For a nonconstant meromorphic function on the unit disc and $q\ge 3$ distinct spherical values, this version of the Nevanlinna second main theorem holds outside a set $E$ with ${\int }_{E}dr/(1-r)<\infty$, as $r\uparrow 1$. The truncated Nevanlinna counting function counts distinct preimages rather than multiplicity. The boundary-distance error is essential and need not be small relative to the characteristic. It follows from the disk Nevanlinna logarithmic derivative lemma and the growth lemma in the variable $\log (1/(1-r))$. The disk error estimate and weighted exceptional-set argument appear in mathweb.tifr.res.in/Documents/Publications/Lectures/tifr17.pdf .

If a transcendental meromorphic function had only finitely many simple preimages over each of five distinct finite values, their simple counting functions would be $O(\log r)$. The displayed multiplicity inequality and the Nevanlinna first main theorem would bound the truncated sum by $5T_f/2+O(\log r)$. The Nevanlinna second main theorem requires at least $3T_f$ up to its $o(T_f)$ error, a contradiction. Thus at least one value has infinitely many simple preimages; four other values can remain totally ramified.

For a nonconstant meromorphic function on the plane, the displayed estimate holds outside an exceptional set of radii of finite linear measure, for large $r$. The logarithmic derivative has only simple principal parts at zeros and poles; Poisson-Jensen formula estimates and control of increasing auxiliary growth functions produce the logarithmic error. It supplies the error term in the Nevanlinna second main theorem.

For a nonnegative increasing continuously differentiable function not identically zero, choose $x_0$ with $\varphi (x_0)>0$. On $I=\{x\ge x_0:{\varphi }^{\prime }>{\varphi }^2\}$, $1<{\varphi }^{\prime }/{\varphi }^2$. Thus $|I|\le {\int }_{x_0}^{\infty }{\varphi }^{\prime }/{\varphi }^2\le 1/\varphi (x_0)$. Adding the finite initial interval controls all exceptions. For the zero function there are none. Taking logarithms turns derivative growth into logarithmic auxiliary-function growth in value-distribution estimates.

The Nevanlinna ramification index is the normalized lower limiting integrated excess multiplicity over a target value, with pole orders used for infinity. It is an asymptotic value-distribution quantity, distinct from the integer ramification index of a holomorphic map at one source point. It can be positive even when the target has positive Nevanlinna deficiency.

For a nonconstant meromorphic function, its deficiency at a point of the Riemann sphere is the displayed asymptotic proportion missing from the full Nevanlinna integrated counting function. The Nevanlinna first main theorem bounds $N(r;a)$ by $T_f(r)+O(1)$ and identifies deficiency with the lower limiting normalized proximity. An omitted value has deficiency one.

The rational composition law for the Nevanlinna characteristic gives $T_f(r)=3r/\pi +O(1)$. Its zeros are precisely $2\pi i\mathbb{Z}$, all of multiplicity two, so $N(r;0)=2r/\pi +O(\log r)$ and $\overline{N}(r;0)=r/\pi +O(\log r)$. Hence both its Nevanlinna deficiency and Nevanlinna ramification index at zero equal one third.

For a fixed finite target $a$, the displayed identity compares growth with proximity to and occurrences of $a$. If $g(z)=c{z}^{\nu }(1+O(z))$ near zero, where $\nu$ is its signed zero order, the exact reciprocal identity is $T(r,g)-T(r,1/g)=\log |c|$. It follows by factoring out zeros and poles on a disc, applying the mean value property for harmonic functions to the remaining nonvanishing factor, and using ${\log }^{+}u-{\log }^{+}(1/u)=\log u$. Translation changes $T$ by a bounded amount, proving the general assertion.

The characteristic of a meromorphic function is the sum of its Nevanlinna proximity function and Nevanlinna integrated counting function. It treats large values and actual poles together. Replacing $f$ by $f-a$ changes the characteristic by at most $\log (1+|a|)$.

The growth order of a meromorphic function is the upper power exponent of its Nevanlinna characteristic, with constants assigned order zero. It is different from the local signed order at a zero or pole. A Nevanlinna integrated counting function has an analogous growth order. The Nevanlinna first main theorem bounds every counting order by the function order.

At most two values can have counting order smaller than a positive finite function order. Otherwise the Nevanlinna second main theorem with three such values bounds the characteristic by a smaller power outside a finite-length exceptional set. Monotonicity extends this bound to all large radii by choosing a good radius in $[r,r+1]$, contradicting the original order. Order zero is immediate from nonnegativity and the first theorem.

For a degree-$d$ rational function $U$ of the Riemann sphere, choose relatively prime homogeneous polynomials $(P,Q)$ of degree $d$. Their joint norm on the unit three-sphere has positive lower and finite upper bounds. Thus $\log \Vert (P,Q)(v)\Vert -d\log \Vert v\Vert$ is bounded. Averaging over a circle using a nonvanishing entire homogeneous lift of a meromorphic function proves the displayed law for the spherical characteristic. Jensen's formula and $0\le \frac{1}{2}\log (1+x^2)-{\log }^{+}x\le \frac{1}{2}\log 2$ show that it also holds for the ordinary Nevanlinna characteristic.

The number $n(t,f)$ counts poles in $|z|<t$ with multiplicity, and $n(0,f)$ is the pole order at zero. Thus $N(r,f)=n(0,f)\log r+{\sum }_{0<|p|<r}\log (r/|p|)$. Applying this to $1/(f-a)$ counts occurrences of the value $a$.

The truncated counting function counts each distinct $a$-point of a meromorphic function once, rather than with its local multiplicity. It uses the same logarithmic radial weights as the Nevanlinna integrated counting function, including an origin contribution $\log r$ when appropriate. The difference $N-\overline{N}$ measures integrated excess multiplicity.

Here ${\log }^{+}x=\max (0,\log x)$. The function $m(r,f)$ measures logarithmic proximity of $f$ to infinity. For a finite target $a$, proximity is $m(r,1/(f-a))$. Logarithmic singularities on a circle are integrable.

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.

A degree-$m$ rational map of the complex projective line is represented by two degree-at-most-$m$ polynomials without a common root, modulo common scalar multiplication. This is an open set of ${\mathbb{CP}}^{2m+1}$. Quotienting by domain Möbius transformations subtracts three complex dimensions, giving real dimension $4m-4$ for degree $m\ge 2$. These are the local covering parameters of a multiple-covered J-holomorphic 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.

A Padé approximant is a rational function with numerator degree at most $m$ and denominator degree at most $n$, normalized by $Q_n(0)=1$, such that $Q_n(z)f(z)-P_m(z)=O(z^{m+n+1})$ near zero, when such an approximant exists. For the exponential function, its $[1/1]$ approximant is $(1+z/2)/(1-z/2)$. Applying this rational function to a matrix gives the Crank-Nicolson method. Diagonal and subdiagonal approximants also give stability functions of Gauss collocation methods and Radau IIA methods respectively.

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 $a$.

For $A\ne 0$ and $m\ne n$, combining the right-hand fractions gives numerator $A(m-n)$ and proves the identity. Integration on an interval avoiding the poles gives $\ln |(A+mx)/(A+nx)|/[A(m-n)]$ plus a constant. The overall $1/A$ cannot be dropped unless the variable has already been normalized to $A=1$.

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.

The local multiplicity in zeros and poles is positive for a zero and negative for a pole; it is zero at a regular nonvanishing point.

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.