Question Number 167419 by Bagus1003 last updated on 16/Mar/22
$$\left(\frac{\mathrm{1}}{\mathrm{2}}\right)!=? \\ $$$${Use}\:{Method}\:{and}\:{Formula}!!!!!!!!! \\ $$
Commented by mkam last updated on 16/Mar/22
$$\left(\frac{\mathrm{1}}{\mathrm{2}}\right)!\:=\:\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:=\frac{\mathrm{1}}{\mathrm{2}}\:\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:=\:\frac{\sqrt{\pi}}{\mathrm{2}} \\ $$
Answered by alephzero last updated on 16/Mar/22
$${n}!\:=\:\Gamma\left({n}+\mathrm{1}\right) \\ $$$$\Rightarrow\:\left(\frac{\mathrm{1}}{\mathrm{2}}\right)!\:=\:\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:=\:\Gamma\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$\Gamma\left({n}\right)\Gamma\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\:=\:\mathrm{2}^{\mathrm{1}−\mathrm{2}{n}} \sqrt{\pi}\:\Gamma\left(\mathrm{2}{n}\right) \\ $$$$\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:=\:\Gamma\left(\mathrm{1}\right)\Gamma\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\right)\:=\:\Gamma\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$=\:\mathrm{2}^{\mathrm{1}−\mathrm{2}} \sqrt{\pi}\:\Gamma\left(\mathrm{2}\right)\:=\:\frac{\sqrt{\pi}}{\mathrm{2}} \\ $$