DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

Fresnel, error, and Gauss hypergeometric series

Definition

For $z \in C$ , define the Fresnel functions and error function by

$C (z) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k} (π / 2)^{2 k} z^{4 k + 1}}{(2 k)! (4 k + 1)}, S (z) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k} (π / 2)^{2 k + 1} z^{4 k + 3}}{(2 k + 1)! (4 k + 3)}, \erf (z) = \frac{2}{\sqrt{π}} \sum_{k = 0}^{\infty} \frac{(- 1)^{k} z^{2 k + 1}}{k! (2 k + 1)}, erfc (z) = 1 - \erf (z) .$

The following local lemmas prove convergence and show that $C, S, \erf$ equal the standard entire integral functions with their usual normalizations.

For $a, b, c, z \in C$ with $c \notin {0, - 1, - 2, \dots}$ and $∣ z ∣ < 1$ , define

$(a)_{0} = 1, (a)_{k} = \prod_{j = 0}^{k - 1} (a + j),_{2} F_{1} (a, b; c; z) = \sum_{k = 0}^{\infty} \frac{(a)_{k} (b)_{k}}{(c)_{k} k!} z^{k} .$

Only this Gauss-series branch of $_{2} F_{1}$ is defined on this page; no value outside $∣ z ∣ < 1$ or across a continuation cut is implicit.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

DLMF, §7.6, Series Expansions (standard reference, not scraped)
DLMF, §15.2(i), Gauss Series (standard reference, not scraped)
N. M. Temme, “Numerical aspects of special functions,” Acta Numerica 17 (2008), §2 (standard reference, not scraped)