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Negative Laurent coefficients vanish on a Hartogs figure

Statement

In the notation of Laurent coefficients on Hartogs slices depend holomorphically on the remaining variables, one has

$a_{n} (w) = 0 for every n < 0 and every ∣ w ∣ < 1.$

Facts & Assumptions

Given: A holomorphic function $f$ on $H (r, s)$ , a radius $ρ$ with $r < ρ < 1$ , and the Laurent coefficient functions $a_{n}$ defined in the preceding lemma.

[L1]

Each $a_{n}$ is holomorphic on the unit disc, and for fixed $w$ the numbers $a_{n} (w)$ are the Laurent coefficients of the slice $z \mapsto f (z, w)$ on the annulus $r < ∣ z ∣ < 1$ (Laurent coefficients on Hartogs slices depend holomorphically on the remaining variables).

[L2]

A holomorphic function on a punctured disc has a removable singularity exactly when its Laurent expansion has no negative powers (Characterizations of removable singularities).

[L3]

A holomorphic function on a connected open set that vanishes on a nonempty open subset vanishes identically (A holomorphic function vanishing on a nonempty open subset of a domain vanishes identically).

Proof

technique · direct

1.1L1L2

Fix $w$ with $∣ w ∣ < s$ . Then $(z, w) \in H (r, s)$ for every $∣ z ∣ < 1$ , so the slice $z \mapsto f (z, w)$ is holomorphic on the whole unit disc. By [L1], the coefficients $a_{n} (w)$ are the Laurent coefficients of that slice on $r < ∣ z ∣ < 1$ , and [L2] therefore forces $a_{n} (w) = 0$ for every $n < 0$ .

2.1step 1.1L3

For each fixed $n < 0$ , step 1.1 shows that the holomorphic function $a_{n}$ is zero on the nonempty open disc ${∣ w ∣ < s}$ . The domain ${∣ w ∣ < 1}$ is connected, so [L3] gives $a_{n} \equiv 0$ there.

3.1step 2.1∎

As $n < 0$ was arbitrary, every negative Laurent coefficient vanishes on the whole parameter disc ${∣ w ∣ < 1}$ .

Depends on

Used by

A holomorphic function on a Hartogs figure extends to the full bidisc Theorem

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Tasty Bits of Several Complex Variables, §2.1 (standard reference, not scraped)