ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-26

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The power series of $z_{0} z_{1}$ on a bidisc centred away from the origin

Example

Let $f (z) = z_{0} z_{1}$ on $C^{2}$ , and expand about the centre $a = (1, 1)$ . Then

$f (z) = 1 + (z_{0} - 1) + (z_{1} - 1) + (z_{0} - 1) (z_{1} - 1) .$

So the multi-indexed power series at $a$ has coefficients $c_{(0, 0)} = c_{(1, 0)} = c_{(0, 1)} = c_{(1, 1)} = 1$ and $c_{α} = 0$ for every other $α \in N^{2}$ . Being finite, the series converges absolutely on every bidisc centred at $(1, 1)$ .

Facts & Assumptions

Given: The function $f (z) = z_{0} z_{1}$ on $C^{2}$ , the centre $a = (1, 1)$ , and the bidisc notation of Balls, polydiscs and the distinguished boundary in $C^{m}$ .

Verification

technique · direct

1.1givenalgebra

Writing $z_{k} = 1 + (z_{k} - 1)$ for $k = 0, 1$ and expanding gives $z_{0} z_{1} = (1 + (z_{0} - 1)) (1 + (z_{1} - 1)) = 1 + (z_{0} - 1) + (z_{1} - 1) + (z_{0} - 1) (z_{1} - 1)$ .

2.1step 1.1

The right-hand side is a finite multi-indexed power series about $(1, 1)$ , with exactly the four nonzero coefficients stated above, so it converges absolutely on every bidisc centred at $(1, 1)$ .

3.1step 2.1algebra∎

The displayed finite series already equals $f$ everywhere, so it is in particular the power-series expansion of $f$ about $(1, 1)$ . Also $\partial^{(1, 0)} f (1, 1) = 1$ , $\partial^{(0, 1)} f (1, 1) = 1$ , and $\partial^{(1, 1)} f (1, 1) = 1$ by direct differentiation, while every derivative of order at least $2$ in one coordinate is $0$ ; these values match the displayed coefficients.

Depends on

Used by

Dependency tree · two levels

47 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, v4.4, §1.2 (standard reference, not scraped)

The power series of z0z1z0​z1​ on a bidisc centred away from the origin