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?
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This question has been asked many, many times on this site before. – Akiva Weinberger Jan 10 '16 at 04:37
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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. – Forever Mozart Jan 10 '16 at 04:38
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6http://math.stackexchange.com/questions/396889/how-to-find-the-factorial-of-a-fraction?rq=1 – kccu Jan 10 '16 at 04:38
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I particularly like my answer to this question, but I think I'm biased. – Akiva Weinberger Jan 10 '16 at 04:42
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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
Ian Miller
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I think you mean $\Gamma(t)$. Also, that identity works only for positive integers. – YoTengoUnLCD Jan 10 '16 at 04:47