PlanetMath: Euler reflection formula

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Euler reflection formula

(Theorem)

Theorem 1 (Euler Reflection Formula)

$\displaystyle \Gamma(x)\Gamma(1-x)=\frac{\pi}{\sin(\pi x)}$

Proof: We have

$\displaystyle \frac{1}{\Gamma(x)}=xe^{\gamma x}\prod_{n=1}^{\infty} \left(\left(1+\frac{x}{n}\right)e^{-x/n}\right)$

and thus

$\displaystyle \frac{1}{\Gamma(x)}\frac{1}{\Gamma(-x)}=-x^2e^{\gamma x}e^{-\gamm... ...{n}\right)e^{x/n}\right)=-x^2\prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2}\right)$

But $\Gamma(1-x)=-x\Gamma(-x)$ and thus

$\displaystyle \frac{1}{\Gamma(x)}\frac{1}{\Gamma(1-x)}=x\prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2}\right)$

Now, using the formula for $\sin x/x$ , we have

$\displaystyle \sin(\pi x)=\pi x\prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2}\right)$

so that

$\displaystyle \frac{1}{\Gamma(x)}\frac{1}{\Gamma(1-x)}=\frac{\sin(\pi x)}{\pi}$

and the result follows.

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This is version 2 of Euler reflection formula, born on 2006-11-11, modified 2006-11-12.
Object id is 8539, canonical name is EulerReflectionFormula.
Accessed 1769 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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