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Orders and residues under inversion on the sphere

Example

On the Riemann sphere $\hat{C}$ with the standard charts $ϕ_{0} (z) = z$ on $U_{0} = \hat{C} ∖ {\infty}$ and $ϕ_{\infty} (z) = 1 / z$ on $U_{\infty} = \hat{C} ∖ {0}$ , consider the meromorphic differentials $ω = \frac{d z}{z}$ and $η = d z$ . Then:

$ω$ has simple poles exactly at $0$ and $\infty$ , with ${Res}_{0} (ω) = + 1$ and ${Res}_{\infty} (ω) = - 1$ ;
$η$ has a pole of order $2$ at $\infty$ — that is, ${ord}_{\infty} (η) = - 2$ — with ${Res}_{\infty} (η) = 0$ , and no other pole;
in both cases the residues sum to $0$ , in agreement with the residue theorem on a compact Riemann surface.

Facts & Assumptions

Given: The Riemann sphere $\hat{C}$ with its standard charts $ϕ_{0}, ϕ_{\infty}$ , and the differentials $ω = d z / z$ and $η = d z$ .

[F1]

$ϕ_{0} : U_{0} \to C$ is $z \mapsto z$ and $ϕ_{\infty} : U_{\infty} \to C$ is $z \mapsto 1 / z$ (with $ϕ_{\infty} (\infty) = 0$ ), where $U_{0} = \hat{C} ∖ {\infty}$ and $U_{\infty} = \hat{C} ∖ {0}$ ; the transitions on the overlap $C^{\times}$ are $w \mapsto 1 / w$ in both directions (The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity).

[F2]

With these charts $\hat{C}$ is a Riemann surface (Atlases on the sphere, plane, disc and annulus), and it is compact and Hausdorff, being the one-point compactification of $C$ (The Riemann sphere is the published one-point compactification of the complex plane).

[F3]

A meromorphic differential on a Riemann surface is a family of local meromorphic expressions $h_{φ}$ satisfying the transition law $h_{ψ} (w) = h_{φ} (z (w)) z^{'} (w)$ on overlaps, where $z = φ \circ ψ^{- 1}$ ; its order at a point is the Laurent order of any centred local expression and its residue is the coefficient of $z^{- 1}$ there (Meromorphic differentials, orders and residues).

[F4]

The residue of an isolated singularity is the Laurent coefficient $c_{- 1}$ , and is unchanged by shrinking the punctured disc; a local expression $z^{- 1}$ has residue $1$ at $0$ , and a local expression with Laurent expansion $c_{- 2} z^{- 2}$ has residue $0$ at $0$ (The residue of an isolated singularity, Isolated singularities: removable, poles, and essential singularities).

[F5]

The reciprocal rule for complex derivatives: if $g (a) \neq 0$ then $(1 / g)^{'} (a) = - g^{'} (a) / g (a)^{2}$ (Linearity, product, reciprocal, and quotient rules for complex derivatives); in particular the map $w \mapsto 1 / w$ has derivative $- 1 / w^{2}$ on $C^{\times}$ , so with $z = 1 / w$ one has $d z = - \frac{d w}{w^{2}}$ .

[F6]

Residue theorem on a compact Riemann surface: a meromorphic differential on a compact Riemann surface has only finitely many nonzero residues and their total sum is $0$ (Residue theorem on a compact Riemann surface).

Verification

technique · direct

1.1F1F3F4

(The finite-chart expression of $ω$ .) In the chart $ϕ_{0}$ the local expression of $ω = \frac{d z}{z}$ is $h_{ϕ_{0}} (z) = 1 / z$ , holomorphic on $C^{\times}$ with Laurent expansion $z^{- 1}$ at $0$ ; hence ${ord}_{0} (ω) = - 1$ and ${Res}_{0} (ω) = 1$ , and $0$ is the only pole of $ω$ in the chart $U_{0}$ .

1.2F1F3F4F5

(The infinity-chart expression of $ω$ .) Put $w = ϕ_{\infty} (z) = 1 / z$ , so that $z = 1 / w$ and $z^{'} (w) = - 1 / w^{2}$ by the reciprocal rule; the transition law gives $h_{ϕ_{\infty}} (w) = h_{ϕ_{0}} (z (w)) z^{'} (w) = w \cdot (- 1 / w^{2}) = - 1 / w$ . Thus the chart expression at infinity has a simple pole at $w = 0$ , the point $z = \infty$ , so ${ord}_{\infty} (ω) = - 1$ and ${Res}_{\infty} (ω) = - 1$ , and it is holomorphic on $U_{\infty} ∖ {\infty}$ .

1.3F1F3F4F5

(The differential $η = d z$ in both charts.) In the chart $ϕ_{0}$ the local expression of $η$ is $h_{ϕ_{0}} \equiv 1$ , holomorphic on all of $C$ , so $η$ has no pole in $U_{0}$ and ${ord}_{p} (η) = 0$ for $p \in C$ ; in the chart $ϕ_{\infty}$ the transition law with the same factor $z^{'} (w) = - 1 / w^{2}$ gives $h_{ϕ_{\infty}} (w) = 1 \cdot (- 1 / w^{2}) = - 1 / w^{2}$ , holomorphic on $C^{\times}$ ; hence ${ord}_{\infty} (η) = - 2$ and ${Res}_{\infty} (η) = 0$ , the Laurent expansion $- w^{- 2}$ having no $w^{- 1}$ term.

2.1F1step 1.1step 1.2

(Pole set of $ω$ .) The two chart domains cover $\hat{C}$ , the only pole of $h_{ϕ_{0}}$ is $z = 0$ and the only pole of $h_{ϕ_{\infty}}$ is $w = 0$ , corresponding to $z = \infty$ ; hence the pole set of $ω$ is exactly ${0, \infty}$ , both poles simple, with residues $+ 1$ and $- 1$ .

2.2step 1.3

(Pole set of $η$ .) Since $h_{ϕ_{0}}$ is holomorphic on $U_{0}$ and the only pole of $h_{ϕ_{\infty}}$ is at $w = 0$ , the pole set of $η$ is exactly ${\infty}$ , a single pole of order $2$ with residue $0$ .

3.1F2F3F6step 2.1step 2.2∎

(Agreement with the residue theorem.) The sphere is a compact Riemann surface [F2], $ω$ and $η$ are meromorphic differentials on it [F3], and each has a finite pole set, so [F6] applies to both; the totals are ${Res}_{0} (ω) + {Res}_{\infty} (ω) = 1 + (- 1) = 0$ and ${Res}_{\infty} (η) = 0$ , and in each case only finitely many residues are nonzero, so both computations agree with the residue theorem.

Remarks

The inversion $z = 1 / w$ is exactly what the example is testing: in the chart at infinity the differential $d z$ becomes $- d w / w^{2}$ , so the expression that looks constant in the finite chart acquires a double pole at infinity, and $d z / z$ , which has residue $+ 1$ at $0$ , acquires residue $- 1$ at $\infty$ because the transition factor $z^{'} (w) = - 1 / w^{2}$ turns the expression $w$ into $- 1 / w$ . This is the transition law of Meromorphic differentials, orders and residues in its simplest nontrivial instance, and it shows that order and residue are genuinely features of the differential and not of a chosen coordinate. The sphere is also the smallest illustration of the residue theorem: a nonzero residue at a finite point must be balanced by a residue elsewhere, and a meromorphic differential whose only pole is at infinity must have residue $0$ there, as $η = d z$ illustrates.

Depends on

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Sources

Eduard Looijenga, Riemann Surfaces (2007) (standard reference, not scraped)
Curtis T. McMullen, Riemann Surfaces, Math 213b course notes (2026) (standard reference, not scraped)