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Laurent coefficients on Hartogs slices depend holomorphically on the remaining variables

Statement

Fix $0 < r, s < 1$ , let $f$ be holomorphic on $H (r, s)$ , and choose $ρ$ with $r < ρ < 1$ . For each integer $n$ and each $w$ with $∣ w ∣ < 1$ , define

$a_{n} (w) : = \frac{1}{2 π i} \int_{∣ ζ ∣ = ρ} \frac{f (ζ, w)}{ζ^{n + 1}} d ζ .$

Then every $a_{n}$ is holomorphic on the unit disc ${∣ w ∣ < 1}$ . Moreover, for each fixed $w$ with $∣ w ∣ < 1$ the slice $z \mapsto f (z, w)$ has Laurent expansion

$f (z, w) = \sum_{n \in Z} a_{n} (w) z^{n} (r < ∣ z ∣ < 1) .$

Facts & Assumptions

Given: Real numbers $0 < r < s < 1$ are not assumed; only $0 < r, s < 1$ , a function $f \in O (H (r, s))$ , and a radius $ρ$ with $r < ρ < 1$ .

[L1]

The Hartogs figure contains every point $(ζ, w)$ with $∣ ζ ∣ = ρ$ and $∣ w ∣ < 1$ (The Hartogs figure H(r,s) and its bidisc hull).

[L2]

A contour integral of a jointly continuous integrand that is holomorphic in the parameter variable defines a holomorphic function of that parameter (A contour integral of a jointly continuous, parameter-holomorphic integrand is holomorphic).

[L3]

The Laurent coefficients of a one-variable holomorphic function on an annulus are given by the contour integral formula, and those coefficients are unique (Laurent coefficients are given by contour integrals and are unique).

Proof

technique · direct

1.1

Fix an integer $n$ . By [L1], for $∣ ζ ∣ = ρ$ and $∣ w ∣ < 1$ the point $(ζ, w)$ lies in $H (r, s)$ , so the integrand $(ζ, w) \mapsto f (ζ, w) ζ^{- n - 1}$ is continuous on the circle times the unit disc and, for fixed $ζ$ , holomorphic in $w$ . Therefore [L2] makes $a_{n}$ holomorphic on ${∣ w ∣ < 1}$ .

L1L2

1.2

Fix $w$ with $∣ w ∣ < 1$ . Then the slice $z \mapsto f (z, w)$ is holomorphic on the annulus $r < ∣ z ∣ < 1$ , again by [L1]. Applying [L3] to that one-variable slice on the circle $∣ z ∣ = ρ$ shows that its Laurent coefficients are exactly the numbers $a_{n} (w)$ defined above.

L1L3

2.1

The Laurent expansion from step 1.2 is therefore $f (z, w) = \sum_{n \in Z} a_{n} (w) z^{n}$ for $r < ∣ z ∣ < 1$ , and step 1.1 gives the holomorphic dependence of every coefficient on $w$ .

step 1.1step 1.2∎

Depends on

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25 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)