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$Γ (1 / 2) = \sqrt{π}$ from the Gaussian integral

Statement

$Γ (1 / 2) = \sqrt{π}$ .

Facts & Assumptions

Given: Euler's Gamma integral at $s = 1 / 2$ .

[F1]

If $ϕ : I \to J$ is a monotone differentiable surjection with locally integrable derivative, the proper change-of-variable hypotheses hold on every compact truncation, and $f$ is locally integrable on $J$ , then the improper integrals of $f$ and $f (ϕ) ∣ ϕ^{'} ∣$ converge simultaneously and are equal when convergent (Change of variable in an improper integral).

[F2]

The Gaussian integral is $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ (The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ ).

Proof

technique · direct

1.1

In $Γ (1 / 2) = \int_{0}^{\infty} t^{- 1 / 2} e^{- t} d t$ , use $t = u^{2}$ on proper truncations. By [F1], the improper limit is $2 \int_{0}^{\infty} e^{- u^{2}} d u$ .

F1algebra

2.1

The integrand is even, so splitting [F2] at zero shows $2 \int_{0}^{\infty} e^{- u^{2}} d u = \sqrt{π}$ .

step 1.1F2algebra

3.1

Combining the two identities gives $Γ (1 / 2) = \sqrt{π}$ , with the positive square root selected because Gamma is positive.

step 2.1∎

Remarks

The independent Wallis-product route is $Γ (1 / 2) = \sqrt{π}$ by Wallis's product.

Depends on

Used by

The closed form for the volume of the unit n-ball Corollary
The unit-ball volume is maximal in dimension five Corollary
Special values of the real Gamma and Beta functions Example
The unit-ball volumes through dimension eight from the Gamma formula Example

Dependency tree · two levels

20 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

University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §1.4 (standard reference, not scraped)

Γ(1/2)=πΓ(1/2)=π​ from the Gaussian integral