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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?

Rohinb97
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1 Answers1

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The Factorial of a Rational number is defined by the Gamma function. A link is in the 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.

Matthew Schmidt
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iostream007
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  • Missing a factor. $\Gamma(0.5)=(-0.5)!=\sqrt{\pi}$. Note your answer is twice the answer given by the calculator. The calculator is correct. – Thomas Andrews Jul 28 '13 at 13:28
  • @ThomasAndrews I've edited my answer.Actually I forget $\frac 12$ that's why my answer was twice than calculator answer – iostream007 Jul 29 '13 at 11:25
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    Once you switch to gamma notation, don't switch back - just use $\Gamma.$ And why use $\implies$ when $=$ is just as good? – Thomas Andrews Jul 29 '13 at 12:27