Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 67631 by Hassen_Timol last updated on 29/Aug/19

$$$$$$$$$$\mathrm{Can}\mathrm{you}\mathrm{please}\mathrm{tell}\mathrm{me},\mathrm{where}\mathrm{does}\mathrm{this}$$$$\mathrm{formula}\mathrm{come}\mathrm{from}?$$$$\mathrm{And}\mathrm{what}\mathrm{means}\mathrm{the}\mathrm{factorial}\mathrm{of}\mathrm{a}\mathrm{non}-$$$$\mathrm{integer}\mathrm{number}?$$$$$$$$\pi ={(\frac{1}{2}!)}^2\times 4$$$$$$$$I^{\prime }veverifiedtheaboveequationwithcalculator.$$$$$$$$\mathrm{Thank}\mathrm{you}$$$$$$$$$$

Commented by mathmax by abdo last updated on 29/Aug/19

$$wehavex!=\mathrm{\Gamma }(x+1)=x\mathrm{\Gamma }\left(x\right)\Rightarrow (\frac{1}{2}!)=\mathrm{\Gamma }(\frac{1}{2}+1)=\frac{1}{2}\mathrm{\Gamma }\left(\frac{1}{2}\right)\Rightarrow$$$${(\frac{1}{2}!)}^2\times 4=\frac{1}{4}{\mathrm{\Gamma }}^2\left(\frac{1}{2}\right)\times 4={\mathrm{\Gamma }}^2\left(\frac{1}{2}\right)but$$$$\mathrm{\Gamma }\left(\frac{1}{2}\right)={\int }_0^{\mathrm{\infty }}{t}^{\frac{1}{2}-1}{e}^{-t}dt={\int }_0^{\mathrm{\infty }}\frac{e^{-t}}{\sqrt{t}}dt{=}_{\sqrt{t}=u}{\int }_0^{\mathrm{\infty }}\frac{e^{-u^2}}{u}\left(2u\right)du$$

Commented by mathmax by abdo last updated on 29/Aug/19

$$\mathrm{\Gamma }\left(\frac{1}{2}\right)=2{\int }_0^{\mathrm{\infty }}{e}^{-u^2}du=2\frac{\sqrt{\pi }}{2}=\sqrt{\pi }\Rightarrow {(\frac{1}{2}!)}^2\times 4={\mathrm{\Gamma }}^2\left(\frac{1}{2}\right)=\pi .$$

Commented by Hassen_Timol last updated on 03/Sep/19

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}.$$$$\mathrm{But}\mathrm{what}\mathrm{means}\mathrm{the}\mathrm{\Gamma }\left(x\right)\mathrm{function}\mathrm{and}\mathrm{how}$$$$\mathrm{is}\mathrm{it}\mathrm{related}\mathrm{to}\mathrm{the}x!\mathrm{function}?$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com