ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-26

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The power series of $z_{0} / (1 - z_{1})$ and the shape of its domain of convergence

Example

On the region ${(z_{0}, z_{1}) \in C^{2} : ∣ z_{1} ∣ < 1}$ one has

$\frac{z_{0}}{1 - z_{1}} = \sum_{j = 0}^{\infty} z_{0} z_{1}^{j} .$

Equivalently, in multi-index notation,

$\frac{z_{0}}{1 - z_{1}} = \sum_{α \in N^{2}} c_{α} z^{α},$

where $c_{(1, j)} = 1$ for every $j \in N$ and $c_{α} = 0$ otherwise. When $z_{0} \neq 0$ , the series converges absolutely exactly when $∣ z_{1} ∣ < 1$ ; when $z_{0} = 0$ , every term vanishes and the series converges absolutely for every $z_{1}$ . Thus its absolute-convergence set is $(C \times D (0, 1)) \cup ({0} \times C)$ , an unbounded set and not a bounded polydisc.

Facts & Assumptions

Given: The function $f (z) = z_{0} / (1 - z_{1})$ on the region $∣ z_{1} ∣ < 1$ .

[L1]

For complex $w$ , the geometric series $\sum_{j \geq 0} w^{j}$ converges absolutely exactly when $∣ w ∣ < 1$ : if $∣ w ∣ < 1$ , then $\sum_{j \geq 0} ∣ w ∣^{j}$ converges by For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges, so Every absolutely convergent complex series converges, and rearrangements preserve its sum applies, and the finite identity $(1 - w) \sum_{j < n} w^{j} = 1 - w^{n}$ together with $∣ w^{n} ∣ = ∣ w ∣^{n} \to 0$ gives the sum $1 / (1 - w)$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive, For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ , $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ ). If $∣ w ∣ \geq 1$ , then the same real geometric-series criterion shows that $\sum_{j \geq 0} ∣ w ∣^{j}$ diverges, so for every nonzero complex constant $c$ the series $\sum_{j \geq 0} c w^{j}$ cannot be absolutely convergent.

Verification

technique · direct

1.1givenL1

If $∣ z_{1} ∣ < 1$ , then [L1] gives $(1 - z_{1})^{- 1} = \sum_{j \geq 0} z_{1}^{j}$ , so multiplying by $z_{0}$ yields $z_{0} / (1 - z_{1}) = \sum_{j \geq 0} z_{0} z_{1}^{j}$ .

2.1step 1.1

In multi-index form this is the stated coefficient rule: the only monomials that appear are $z_{0} z_{1}^{j}$ , so $c_{(1, j)} = 1$ and every other coefficient is $0$ .

2.2step 1.1L1

The absolute-value series is $\sum_{j \geq 0} ∣ z_{0} ∣ ∣ z_{1} ∣^{j}$ . If $z_{0} \neq 0$ , division by the positive constant $∣ z_{0} ∣$ and [L1] show that it converges exactly when $∣ z_{1} ∣ < 1$ . If $z_{0} = 0$ , every term is $0$ , so it converges for every $z_{1}$ . Hence the absolute-convergence set is $(C \times D (0, 1)) \cup ({0} \times C)$ .

3.1step 2.2∎

At points $(0, z_{1})$ with $∣ z_{1} ∣ \geq 1$ , every term of the series is $0$ , so the series still converges there to $0$ , although the quotient is undefined when $z_{1} = 1$ and this exceptional convergence set is not open.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

74 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 z0/(1−z1)z0​/(1−z1​) and the shape of its domain of convergence