CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The value of Gamma at one half

Statement

$Γ (1 / 2) = \sqrt{π} .$

Facts & Assumptions

Given: The reflection formula and the real Gamma value.

[L1]

On the positive real axis, complex Gamma agrees with the real Gamma (The complex Gamma function restricts to the real Gamma function).

[L2]

The real Gamma function satisfies $Γ (1 / 2) = \sqrt{π}$ ( $Γ (1 / 2) = \sqrt{π}$ from the Gaussian integral).

[L3]

The reflection formula gives $Γ (z) Γ (1 - z) = π / \sin (π z)$ (Euler's reflection formula).

Proof

technique · direct

1.1L3given

Substituting $z = 1 / 2$ into [L3] gives $Γ (1 / 2)^{2} = π$ , because $\sin (π / 2) = 1$ .

2.1step 1.1L1L2∎

By [L1] and [L2], $Γ (1 / 2)$ is the positive real number $\sqrt{π}$ . Step 1.1 leaves only the two square roots of $π$ , so the positive one is the required value.

Depends on

Used by

B(1/2,1/2)=π Example
Checking the reflection formula at z=1/2 Example
Half-integer and negative-half-integer values of Gamma Example
K-type eigenvalues of A(nu): recurrence, closed form and nonvanishing Lemma

Dependency tree · two levels

13 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

K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 6 §1(vii) (standard reference, not scraped)