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A continuous separately holomorphic function is the sum of an absolutely convergent power series with Cauchy-integral coefficients on every smaller polydisc

Statement

Fix $m \geq 1$ , $a \in C^{m}$ and a polyradius $ρ$ , let $f : Δ_{ρ} (a) \to C$ be continuous and separately holomorphic, let $r$ be a polyradius with $r_{k} < ρ_{k}$ for every $k < m$ , and let $C_{k} (t) = a_{k} + r_{k} \exp (i t)$ on $[0, 2 π]$ . For each multi-index $α$ set

$c_{α} : = \frac{1}{(2 π i)^{m}} \int_{C_{0}} ⁣ \dots ⁣ \int_{C_{m - 1}} f (ζ) \prod_{k < m} (ζ_{k} - a_{k})^{- α_{k} - 1} d ζ_{m - 1} \dots d ζ_{0},$

an iterated integral as in the polydisc Cauchy formula. Then, with $M = \sup_{Γ_{r} (a)} ∣ f ∣$ ,

$∣ c_{α} ∣ \leq M \prod_{k < m} r_{k}^{- α_{k}},$

and for every real $θ$ with $0 < θ < 1$ the series $\sum_{α} c_{α} (z - a)^{α}$ converges absolutely and uniformly on ${\overline{Δ}}_{θ r} (a)$ with

$f (z) = \sum_{α} c_{α} (z - a)^{α} (z \in {\overline{Δ}}_{θ r} (a)) .$

Since every $z \in Δ_{r} (a)$ lies in ${\overline{Δ}}_{θ r} (a)$ for some $θ < 1$ , the expansion holds throughout $Δ_{r} (a)$ .

The coefficients are asserted here only as those iterated integrals. That $c_{α}$ equals $\partial^{α} f (a) / α!$ needs termwise differentiation and is not claimed by this statement.

Facts & Assumptions

Given: The data above, with $C^{m}$ read through Complex $m$ -space and its real coordinate dictionary and $f$ continuous and separately holomorphic on $Δ_{ρ} (a)$ (Separately holomorphic functions).

[L1]

Under these hypotheses, $f (z) = (2 π i)^{- m} \int_{C_{0}} \dots \int_{C_{m - 1}} f (ζ) \prod_{k < m} (ζ_{k} - z_{k})^{- 1} d ζ_{m - 1} \dots d ζ_{0}$ for every $z \in Δ_{r} (a)$ , as an iterated integral each of whose integrands is continuous on its circle (The iterated Cauchy integral formula on a polydisc).

[L2]

For $ζ \in Γ_{r} (a)$ and $z \in {\overline{Δ}}_{θ r} (a)$ with $0 \leq θ < 1$ , $\prod_{k < m} (ζ_{k} - z_{k})^{- 1} = \sum_{α} (z - a)^{α} \prod_{k < m} (ζ_{k} - a_{k})^{- α_{k} - 1}$ , with each term dominated by $\prod_{k < m} θ^{α_{k}} / r_{k}$ and the convergence absolute and uniform in the pair (The Cauchy kernel expands as an absolutely and uniformly convergent multi-indexed geometric series).

[L3]

A multi-indexed series converges absolutely at $z$ when the series along one, equivalently every, enumeration of $N^{m}$ converges absolutely; its sum is independent of the enumeration; and its box partial sums over $B_{N} = {α : α_{k} \leq N}$ converge to that sum (Multi-indexed power series in $C^{m}$ and their absolute convergence).

[L4]

If continuous functions on the trace of a fixed rectifiable contour converge uniformly to a continuous function, their integrals converge to its integral (A uniformly convergent sequence of continuous integrands on a fixed contour permits passage of the limit through the complex line integral).

[L5]

If $∣ g ∣ \leq K$ on the trace of a rectifiable contour $γ$ , with $K \geq 0$ , then $∣ \int_{γ} g d z ∣ \leq K L (γ)$ (ML estimate: a contour integral is bounded by a supremum bound times path length); complex line integrals are linear in the integrand (Complex line integrals are linear in the integrand) and exist for continuous integrands (Continuous integrands have complex and absolute line integrals along every rectifiable path).

[L6]

An absolutely convergent complex series converges and every rearrangement has the same sum (Every absolutely convergent complex series converges, and rearrangements preserve its sum); a dominated series with summable bounds converges absolutely and uniformly (Weierstrass M-test for complex-valued function series); a nonnegative series converges exactly when its partial sums are bounded (A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum); for $∣ r^{'} ∣ < 1$ , $\sum_{k} r^{' k} = 1 / (1 - r^{'})$ (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges).

[L7]

If a property holds at $0$ and passes from $q$ to $q + 1$ , it holds for every natural number (The principle of mathematical induction).

[L8]

Negative integer powers are defined exactly for nonzero complex bases (Integer powers in the complex field).

[L9]

$Δ_{r} (a)$ , ${\overline{Δ}}_{r} (a)$ and $Γ_{r} (a)$ are defined coordinatewise by $∣ z_{k} - a_{k} ∣ < r_{k}$ , $\leq r_{k}$ and $= r_{k}$ (Balls, polydiscs and the distinguished boundary in $C^{m}$ ).

[L10]

The once-traversed circle of radius $r^{'} > 0$ has length $2 π r^{'}$ (Every circle has circumference 2 pi r and circumference-to-diameter ratio pi).

[L11]

A subset of $R^{n}$ is compact exactly when it is closed and bounded (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line); the continuous image of a compact subset is compact (The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset); a compact subset is closed and bounded (A compact subset of a metric space is closed and bounded).

[L12]

$∣ z w ∣ = ∣ z ∣ ∣ w ∣$ and $∣ z + w ∣ \leq ∣ z ∣ + ∣ w ∣$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive); finite sums are additive, scale and are monotone in their terms (Laws of finite sums and finite products).

Proof

technique · direct

1.1

$Γ_{r} (a)$ is closed and bounded in $C^{m}$ , hence compact by [L11] and [L9], and it lies in $Δ_{ρ} (a)$ because $r_{k} < ρ_{k}$ ; so $f$ is continuous on it and $M = \sup_{Γ_{r} (a)} ∣ f ∣$ is a real number by [L11].

givenL9L11

1.2

An induction on the number of remaining integrations ([L7]) using [L5] and [L10] gives the iterated bound: if $∣ h ∣ \leq K$ at every point of $Γ_{r} (a)$ , then the modulus of the iterated integral $\int_{C_{0}} \dots \int_{C_{m - 1}} h d ζ_{m - 1} \dots d ζ_{0}$ is at most $K \prod_{k < m} (2 π r_{k})$ , each step contributing one factor $L (C_{k}) = 2 π r_{k}$ .

givenL5L7L10

2.1

Applying step 1.2 to the integrand of $c_{α}$ , whose modulus on $Γ_{r} (a)$ is at most $M \prod_{k < m} r_{k}^{- α_{k} - 1}$ by [L9] and [L12], gives $∣ c_{α} ∣ \leq (2 π)^{- m} M \prod_{k < m} r_{k}^{- α_{k} - 1} \prod_{k < m} (2 π r_{k}) = M \prod_{k < m} r_{k}^{- α_{k}}$ .

step 1.1step 1.2L8L9L12

2.2

Write $S_{N} (ζ, z)$ for the box partial sum of the expansion in [L2]. It is a finite sum, so multiplying by $f (ζ)$ and integrating iteratedly, [L5] and [L12] give $(2 π i)^{- m} \int_{C_{0}} \dots \int_{C_{m - 1}} f (ζ) S_{N} (ζ, z) d ζ_{m - 1} \dots d ζ_{0} = \sum_{α \in B_{N}} c_{α} (z - a)^{α}$ .

step 1.1L2L5L12

3.1

Fix $θ$ with $0 < θ < 1$ and $z \in {\overline{Δ}}_{θ r} (a)$ . By step 2.1 and [L9], $∣ c_{α} (z - a)^{α} ∣ \leq M \prod_{k < m} θ^{α_{k}}$ , and the box sums of the right-hand side are $M \prod_{k < m} \sum_{j \leq N} θ^{j} \leq M \prod_{k < m} (1 - θ)^{- 1}$ by [L12] and [L6]; every finite subset of $N^{m}$ lies in a box, so [L6] makes $\sum_{α} M \prod_{k} θ^{α_{k}}$ convergent and the M-test gives absolute and uniform convergence of $\sum_{α} c_{α} (z - a)^{α}$ on ${\overline{Δ}}_{θ r} (a)$ .

step 2.1L6L9L12

3.2

By [L2] the difference $S_{N} (ζ, z) - \prod_{k < m} (ζ_{k} - z_{k})^{- 1}$ tends to $0$ uniformly for $ζ \in Γ_{r} (a)$ , so multiplying by $f (ζ)$ and using step 1.1 the products differ by at most $M ε_{N}$ with $ε_{N} \to 0$ ; step 1.2 then bounds the difference of the two iterated integrals by $M ε_{N} \prod_{k < m} (2 π r_{k}) / (2 π)^{m}$ , which tends to $0$ . Hence $\sum_{α \in B_{N}} c_{α} (z - a)^{α}$ converges to the iterated integral of [L1], which is $f (z)$ .

step 1.1step 1.2step 2.2L1L2L4L12

4.1

By step 3.1 and [L3] the box partial sums also converge to the sum $\sum_{α} c_{α} (z - a)^{α}$ ; comparing with step 3.2 gives $f (z) = \sum_{α} c_{α} (z - a)^{α}$ for every $z \in {\overline{Δ}}_{θ r} (a)$ . Since a point of $Δ_{r} (a)$ has $∣ z_{k} - a_{k} ∣ < r_{k}$ for each $k$ , it lies in ${\overline{Δ}}_{θ r} (a)$ for any $θ < 1$ exceeding every $∣ z_{k} - a_{k} ∣ / r_{k}$ , so the expansion holds on all of $Δ_{r} (a)$ .

step 3.1step 3.2L3L9∎

Depends on

Used by

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129 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, §1.2 (standard reference, not scraped)
M. Jabbari, Notes for Analysis and Geometry of Several Complex Variables, §3.1 (standard reference, not scraped)