LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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 Airy Ai function has a negative real zero

Statement

The real Airy function $Ai$ has at least one zero on the negative real axis.

Facts & Assumptions

Given: The Airy series in Airy initial-value series.

[F1]

Airy coefficient recurrence and geometric tail bounds: $Ai$ is entire and satisfies $y^{''} = z y$ .

[F2]

Airy initial-value series: $Ai (0) = 1 / (3^{2 / 3} Γ (2 / 3))$ .

[F3]

Euler's Gamma function on the right half-plane: for real $a > 0$ , $Γ (a) = \int_{0}^{\infty} t^{a - 1} e^{- t} d t$ .

[F4]

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , and The derivatives of sine and cosine are cosine and minus sine supply the chain, product, and sine and cosine derivative formulas used in the real-variable argument.

Proof

1.1F1F2F3F4algebra

By [F3], $Γ (2 / 3) > 0$ , so [F2] gives $Ai (0) > 0$ ; in particular, $Ai$ is not the zero function. Put $u (t) = Ai (- t)$ . By [F1] and the chain rule in [F4], $u^{''} (t) + t u (t) = 0$ for real $t$ .

2.1givenF4step 1.1algebra

Suppose $u$ has no zero on $[1, 1 + π]$ . Continuity gives a constant sign; replace $u$ by $- u$ if necessary so that $u > 0$ there. Set $W (t) = u^{'} (t) \sin (t - 1) - u (t) \cos (t - 1)$ . The product rule and sine and cosine derivatives in [F4], together with the equation in step 1.1, give $W^{'} (t) = (1 - t) u (t) \sin (t - 1) \leq 0$ on this interval.

3.1step 2.1F4contradiction∎

But $W (1) = - u (1) < 0$ and $W (1 + π) = u (1 + π) > 0$ , contradicting that $W$ is nonincreasing. Therefore $u$ has a zero in $[1, 1 + π]$ , and $Ai$ has a zero in $[- (1 + π), - 1]$ .

Depends on

Used by

Relative error becomes unavailable at an Airy zero Example

Dependency tree · two levels

21 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

DLMF, §9.9(i), real zeros of Airy functions (standard reference, not scraped)
DLMF, §§9.2(i)–(ii) and 9.9(i) (standard reference, not scraped)
N. M. Temme, “Numerical aspects of special functions,” Acta Numerica 17 (2008), §2 (standard reference, not scraped)