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Question Number 109895 by qwerty111 last updated on 26/Aug/20

Commented by mathdave last updated on 26/Aug/20

$$thequestiondoesnotconverge$$

Commented by Her_Majesty last updated on 26/Aug/20

$$sorrybut\mathrm{\Gamma }\left(x\right)iscontinuous,ithadbeenthe$$$$mainreasonforits‘‘{birth}^{″}.itisnotdefined$$$$onlyforx\in {\mathbb{Z}}^{-}$$

Commented by Her_Majesty last updated on 26/Aug/20

$$\mathrm{\Gamma }\left(x\right)isdefinedforx\in \mathbb{R}\mathrm{\setminus }{\mathbb{Z}}^{-}$$$$ofcoursewecanonlyapproximateinmost$$$$cases.but{Euler}^{\prime }sintentionwastoexpand$$$$thedefinitionofx!forx\in \mathbb{R}$$$$wecanonlygiveexactvaluesof\mathrm{\Gamma }\left(x\right)for$$$$somex\in \mathbb{Q}$$

Answered by Her_Majesty last updated on 26/Aug/20

$$wehadthismanytimesbefore$$$$\left(1\right)x!isdefinedforx\in \mathbb{N}\Rightarrow nointegralexists$$$$\left(2\right)ifweuse(x-1)!=\mathrm{\Gamma }\left(x\right)wehave$$$$\underset{1}{\stackrel{2}{\int }}\mathrm{\Gamma }\left(x\right)dx\approx .922745950681$$$$noexactvaluepossible$$

Commented by 1549442205PVT last updated on 27/Aug/20

$$\mathrm{Thank}\mathrm{Sir}\mathrm{a}\mathrm{lot}\mathrm{for}{\mathrm{Sir}}^{\prime }\mathrm{s}\mathrm{zeal}$$

Commented by 1549442205PVT last updated on 26/Aug/20

$$\mathrm{Thank}\mathrm{Sir}\mathrm{a}\mathrm{lot}.\mathrm{But}\mathrm{i}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{know}$$$$\mathrm{how}\mathrm{to}\mathrm{calculate}\mathrm{\Gamma }\left(1/\sqrt{2}\right),\mathrm{\Gamma }\left(1/\sqrt{5}\right).\mathrm{If}$$$${\mathrm{don}}^{\prime }\mathrm{t}\mathrm{calculate}\mathrm{that}\mathrm{values}\mathrm{then}$$$$\mathrm{means}\mathrm{it}{\mathrm{isn}}^{\prime }\mathrm{t}\mathrm{defined}\mathrm{on}\mathrm{set}\mathrm{of}$$$$\mathrm{irrational}\mathrm{numbers}\mathrm{which}\mathrm{if}\mathrm{such}\mathrm{as}$$$$\mathrm{it}\mathrm{is}\mathrm{is}\mathrm{interrupted}\mathrm{anywhere},\mathrm{so}$$$${\mathrm{can}}^{\prime }\mathrm{t}\mathrm{have}\mathrm{its}\mathrm{defined}\mathrm{integration}$$$$\mathrm{because}\mathrm{on}\mathrm{segment}[0,1]\mathrm{it}\mathrm{has}\mathrm{an}$$$$\mathrm{infinite}\mathrm{number}\mathrm{of}\mathrm{interrupted}\mathrm{points}$$

Commented by Her_Majesty last updated on 26/Aug/20

you are wrong; read this https://en.wikipedia.org/wiki/Gamma_function

Commented by Her_Majesty last updated on 26/Aug/20

$$\mathrm{\Gamma }\left(1/\sqrt{2}\right)\approx 1.28694088$$$$\mathrm{\Gamma }\left(1/\sqrt{5}\right)\approx 1.98046902$$

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