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How do we find factorial of fractions? For eg: $\frac{1!}{2!}=(\frac{\pi}{4})^{\frac{1}{2}}$ Factorials are used in combinatorics and they can only be functioned on integers to give integers.Then how do we get this?


marked as duplicate by Blue, Akiva Weinberger, Brevan Ellefsen, Ian Miller, JimmyK4542 Jan 10 '16 at 4:59

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  • $\begingroup$ This question has been asked many, many times on this site before. $\endgroup$ – Akiva Weinberger Jan 10 '16 at 4:37
  • $\begingroup$ the factorial function has an extension from $\mathbb N$ to $\mathbb R$ by Tietze. There is a unique such function which is logarithmically convex. Think about it. $\endgroup$ – Forever Mozart Jan 10 '16 at 4:38
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    $\begingroup$ math.stackexchange.com/questions/396889/… $\endgroup$ – kccu Jan 10 '16 at 4:38
  • $\begingroup$ I particularly like my answer to this question, but I think I'm biased. $\endgroup$ – Akiva Weinberger Jan 10 '16 at 4:42

Related to factorials is the gamma $(\Gamma)$ function. For positve integers it satisfies: $\Gamma(n)=(n-1)!$. For non-integers it is given by: $\Gamma(t)=\int_0^\infty x^{t-1}e^{-x}dx$.

Further reading: Wikipedia

  • $\begingroup$ I think you mean $\Gamma(t)$. Also, that identity works only for positive integers. $\endgroup$ – YoTengoUnLCD Jan 10 '16 at 4:47

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