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evaluating the gamma function at 1/2
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(Derivation)
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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,
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 $\Gamma(1/2) = \sqrt{\pi}$ .
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"evaluating the gamma function at 1/2" is owned by CWoo. [ owner history (1) ]
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Cross-references: even function, area under the bell curve, proof, gamma function
There is 1 reference to this entry.
This is version 1 of evaluating the gamma function at 1/2, born on 2007-04-19.
Object id is 9223, canonical name is EvaluatingTheGammaFunctionAt12.
Accessed 1558 times total.
Classification:
| AMS MSC: | 33B15 (Special functions :: Elementary classical functions :: Gamma, beta and polygamma functions) | | | 30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory) |
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Pending Errata and Addenda
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