How do we calculate factorials for numbers with decimal places? [duplicate]

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I was playing with my calculator when I tried $1.5!$. It came out to be $1.32934038817$.

Now my question is that isn't factorial for natural numbers only? Like $2!$ is $2\times1$, but how do we express $1.5!$ like this?

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1 Answer

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factorial of fraction number are defind by gamma function as link is in comments

since

$n!=n\times (n-1)!$

$\Gamma(n)=(n-1)!$

$n!=n \cdot \Gamma(n)$

$\Gamma \left(\dfrac 12\right)=\sqrt\pi$

so$$1.5!= \left(\dfrac 32\right)!= \left(\dfrac 32\right) \cdot \left(\dfrac 12\right)!= \left(\dfrac 32\right) \cdot \left(\dfrac 12\right) \cdot \Gamma{\left(\dfrac 12\right)} = \dfrac 34 \sqrt \pi$$

this can be useful.

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