LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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The binomial theorem over the complex field

Statement

For $z,w\in\mathbb C$ and $n\in\mathbb N$ , write $\iota_{\mathbb C}:\mathbb N\to\mathbb C$ for the canonical-natural map of The canonical natural $\iota(n) = n \cdot 1_F$ of a field. Then $(z+w)^n=\sum_{k\le n}\iota_{\mathbb C}\!\left(\binom nk\right) z^kw^{n-k}.$ The conventions and prerequisite facts used below are recorded in The complex numbers form a field, and every nonzero $x+iy$ has inverse $(x-iy)/(x^2+y^2)$ , Integer powers in the complex field, The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$ , Pascal's rule $\binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}$ , and the hockey-stick identity $\sum_{i \le n}\binom{i}{k} = \binom{n+1}{k+1}$ , The product $g_0 g_1 \cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ( $n = 0$ ) equal to the identity, Generalised associativity: in a monoid the product of a finite list does not depend on the bracketing, and in a commutative monoid it does not depend on the order of the factors either, The principle of mathematical induction.

Facts & Assumptions

Given: Complex $z,w$ and natural $n$ .

Proof

technique · induction

1.1

For $n=0$ both sides are the empty-sum convention $1$ .

base

1.2

Assume the formula at $n$ .

2.1

Multiply the formula in step 1.2 by $z+w$ , split the finite initial-segment sums, prove the shift $k\mapsto k+1$ from the recursive monoid-sum clauses, and group equal powers.

given

3.1

Pascal's rule gives the complex coefficient $\iota_{\mathbb C}\!\left(\binom{n+1}k\right)$ at every index, including the endpoints, so the formula holds at $n+1$ .

discharge-induction∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 84 results over 24 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)