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

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.

Euler's formula: $\exp(i\theta)=\cos\theta+i\sin\theta$ for every real $\theta$

Statement

For every real $\theta$ , $\exp(i\theta)=\cos\theta+i\sin\theta$ .

Facts & Assumptions

Given: A real $\theta$ .

[L1]

The complex exponential by its power series defines $\exp z$ as the sum of the complex series with terms $z^n/n!$ , where the factorial is embedded in the complex field.

[L2]

The complex exponential series converges absolutely for every complex argument states that for every $z\in\mathbb C$ , the series $\sum z^n/n!$ converges absolutely.

[L3]

If $0 \le a_k \le b_k$ eventually, convergence of $\sum b_k$ gives convergence of $\sum a_k$ , and divergence of $\sum a_k$ gives divergence of $\sum b_k$ states that if $0\le u_n\le v_n$ eventually and $\sum v_n$ converges, then $\sum u_n$ converges.

[L4]

Every absolutely convergent complex series converges, and rearrangements preserve its sum states that every absolutely convergent complex series converges.

[L5]

For $n \ge 1$ a sequence in $\mathbb{R}^n$ converges iff each coordinate sequence converges, is Cauchy iff each coordinate sequence is Cauchy, and $\mathbb{R}^n$ is complete in every norm states that a sequence in $\mathbb R^2$ converges exactly when each coordinate sequence converges.

[L6]

Convergent series add and scale termwise states that convergent real series add and scale termwise, with the corresponding sums.

[L7]

Grouping: if $\sum a_k$ converges and $(n_j)$ is strictly increasing with $n_0 = 0$ , the series of blocks $\sum_{k=n_j}^{n_{j+1}-1} a_k$ converges to the same sum states that a convergent real series may be grouped into consecutive finite blocks without changing its sum.

[L8]

Sine and cosine defined by their real power series defines sine and cosine by the real series $\sum_{k=0}^{\infty}(-1)^k\theta^{2k+1}/(2k+1)!$ and $\sum_{k=0}^{\infty}(-1)^k\theta^{2k}/(2k)!$ .

[L9]

Integer powers in the complex field defines natural complex powers by $z^0=1$ and $z^{n+1}=z^nz$ .

Proof

technique · direct

1.1

Put $a_n=(i\theta)^n/n!$ , and define the parity masks by $e_n=a_n$ for even $n$ and $e_n=0$ for odd $n$ , while $o_n=0$ for even $n$ and $o_n=a_n$ for odd $n$ . By [L1] and [L2], $\sum a_n$ converges absolutely and has sum $\exp(i\theta)$ .

L1L2construct

1.2

Induction from $i^2=-1$ and the recursion in [L9] gives $i^{2k}=(-1)^k$ and $i^{2k+1}=i(-1)^k$ for every $k\in\mathbb N$ . Hence $e_{2k}=(-1)^k\theta^{2k}/(2k)!$ , $e_{2k+1}=0$ , $o_{2k}=0$ , and $o_{2k+1}=i(-1)^k\theta^{2k+1}/(2k+1)!$ .

L9algebra

2.1

Since $0\le|e_n|\le|a_n|$ and $0\le|o_n|\le|a_n|$ , [L3] shows that both modulus series converge. Thus $\sum e_n$ and $\sum o_n$ are absolutely convergent and hence converge by [L4].

L3L4step 1.1

3.1

By [L5], the two coordinate series of $e$ and $o$ converge. Since $a_n=e_n+o_n$ , applying [L6] in each coordinate gives $\sum_{n=0}^{\infty}a_n=\sum_{n=0}^{\infty}e_n+\sum_{n=0}^{\infty}o_n$ .

L5L6step 2.1

3.2

Apply [L7] to each real coordinate series of $e$ , with consecutive blocks $[2k,2k+2)$ . By step 1.2 the real-coordinate blocks are $(-1)^k\theta^{2k}/(2k)!$ and the imaginary-coordinate blocks are $0$ . Therefore [L8] identifies $\sum e_n=\cos\theta$ .

L5L7L8step 2.1step 1.2

3.3

Apply the same coordinatewise grouping to $o$ . Its real-coordinate blocks are $0$ and its imaginary-coordinate blocks are $(-1)^k\theta^{2k+1}/(2k+1)!$ , so [L8] gives $\sum o_n=i\sin\theta$ .

L5L7L8step 2.1step 1.2

4.1

Substitute steps 3.2 and 3.3 into step 3.1 and use step 1.1: $\exp(i\theta)=\cos\theta+i\sin\theta$ .

step 1.1step 3.1step 3.2step 3.3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 162 results over 27 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)

Euler's formula: exp⁡(iθ)=cos⁡θ+isin⁡θ\exp(i\theta)=\cos\theta+i\sin\thetaexp(iθ)=cosθ+isinθ for every real θ\thetaθ