## You are here

Homeevaluating the gamma function at 1/2

## Primary tabs

# evaluating the gamma function at 1/2

In the entry on the gamma function it is mentioned that $\Gamma(1/2)=\sqrt{\pi}$. In this entry we reduce the proof of this claim to the problem of computing the area under the bell curve. First note that by definition of the gamma function,

$\displaystyle\Gamma(1/2)$ | $\displaystyle=\int_{0}^{{\infty}}e^{{-x}}x^{{-1/2}}\,dx$ | ||

$\displaystyle=2\int_{0}^{{\infty}}e^{{-x}}\frac{1}{2\sqrt{x}}\,dx.$ |

Performing the substitution $u=\sqrt{x}$, we find that $du=\frac{1}{2\sqrt{x}}\,dx$, so

$\Gamma(1/2)=2\int_{0}^{{\infty}}e^{{-u^{2}}}\,du=\int_{{\mathbb{R}}}e^{{-u^{2}% }}\,du,$ |

where the last equality holds because $e^{{-u^{2}}}$ is an even function. Since the area under the bell curve is $\sqrt{\pi}$, it follows that $\Gamma(1/2)=\sqrt{\pi}$.

Related:

AreaUnderGaussianCurve, LaplaceTransformOfPowerFunction

Type of Math Object:

Derivation

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

30D30*no label found*33B15

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Apr 22

new question: Prove that for any sets A, B, and C, An(BUC)=(AnB)U(AnC) by St_Louis

Apr 20

new image: information-theoretic-distributed-measurement-dds.png by rspuzio

new image: information-theoretic-distributed-measurement-4.2 by rspuzio

new image: information-theoretic-distributed-measurement-4.1 by rspuzio

new image: information-theoretic-distributed-measurement-3.2 by rspuzio

new image: information-theoretic-distributed-measurement-3.1 by rspuzio

new image: information-theoretic-distributed-measurement-2.1 by rspuzio

Apr 19

new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new question: Prove that for any sets A, B, and C, An(BUC)=(AnB)U(AnC) by St_Louis

Apr 20

new image: information-theoretic-distributed-measurement-dds.png by rspuzio

new image: information-theoretic-distributed-measurement-4.2 by rspuzio

new image: information-theoretic-distributed-measurement-4.1 by rspuzio

new image: information-theoretic-distributed-measurement-3.2 by rspuzio

new image: information-theoretic-distributed-measurement-3.1 by rspuzio

new image: information-theoretic-distributed-measurement-2.1 by rspuzio

Apr 19

new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia