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

((1/2))!=? Use Method and Formula!!!!!!!!!

$$\left(\frac{\mathrm{1}}{\mathrm{2}}\right)!=? \\ $$$${Use}\:{Method}\:{and}\:{Formula}!!!!!!!!! \\ $$

Commented by mkam last updated on 16/Mar/22

((1/2))! = Γ((3/2)) =(1/2) Γ((1/2)) = ((√π)/2)

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

Answered by alephzero last updated on 16/Mar/22

n! = Γ(n+1) ⇒ ((1/2))! = Γ((3/2)) = Γ(1+(1/2)) Γ(n)Γ(n+(1/2)) = 2^(1−2n) (√π) Γ(2n) Γ((3/2)) = Γ(1)Γ(1+(1/2)) = Γ(1+(1/2)) = 2^(1−2) (√π) Γ(2) = ((√π)/2)

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

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