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i'm looking for the best method to demonstrate the formula of supplements: $$ \Gamma(x)\Gamma(1-x)=\frac{\pi}{\sin(\pi x)}, \quad \forall x>0$$ in your opinion, what is the best and sample method ! I used a residue calculation but it was very tedious! thank you in advance !

H_K
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  • Proof methods depend on the definition of $\Gamma$ you are working with, in any case you may have a look at Chapter 6 of my notes – Jack D'Aurizio Jun 11 '20 at 08:59
  • In complex analysis it is immediate that $\Gamma(z)\Gamma(1-z)-\frac{\pi}{\sin(\pi z)}$ is entire, $2$-periodic, odd, and bounded on $\Re(z)\in [0,1]$, whence it is entire and bounded thus constant, the constant $0$ is found from $z=i\infty$. – reuns Jun 11 '20 at 09:07
  • Thank you, 1. i used two definitions of $\Gamma$ (Euler product and intergal representation ) ! 2. can you please elaborate ! – H_K Jun 11 '20 at 09:25

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Yes !

First we use Euler product of $sin$ :

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

Then use the Gamma development of Weistrass :

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

Where $\gamma$ denotes the Euler constant.

Now could you end the proof ?

EDX
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