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Question Number 167419 by Bagus1003 last updated on 16/Mar/22

$$\left(\frac{1}{2}\right)!=?$$$$UseMethodandFormula!!!!!!!!!$$

Commented by mkam last updated on 16/Mar/22

$$\left(\frac{1}{2}\right)!=\mathrm{\Gamma }\left(\frac{3}{2}\right)=\frac{1}{2}\mathrm{\Gamma }\left(\frac{1}{2}\right)=\frac{\sqrt{\pi }}{2}$$

Answered by alephzero last updated on 16/Mar/22

$$n!=\mathrm{\Gamma }(n+1)$$$$\Rightarrow \left(\frac{1}{2}\right)!=\mathrm{\Gamma }\left(\frac{3}{2}\right)=\mathrm{\Gamma }(1+\frac{1}{2})$$$$\mathrm{\Gamma }\left(n\right)\mathrm{\Gamma }(n+\frac{1}{2})=2^{1-2n}\sqrt{\pi }\mathrm{\Gamma }\left(2n\right)$$$$\mathrm{\Gamma }\left(\frac{3}{2}\right)=\mathrm{\Gamma }\left(1\right)\mathrm{\Gamma }(1+\frac{1}{2})=\mathrm{\Gamma }(1+\frac{1}{2})$$$$=2^{1-2}\sqrt{\pi }\mathrm{\Gamma }\left(2\right)=\frac{\sqrt{\pi }}{2}$$

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