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I was always taught that $n!$ simple meant multiplying together all numbers up to and including $n$. So for example, $3!$ meant $3 \times 2 \times 1$. Now I discover that $n$ does not have to be an integer. SpeedCrunch, the Linux calculator app will calculate $3.5!$ as $11.6$. So how would I write that down long handed?

To be clear, how would I write down 3.5! as 1 x 2 x 3 x (???). The so called duplicate question does not address this specific issue. Plus I can't understand any of the complex math there; I'm not a mathematician. Can anyone think of a simpler answer?

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See the Gamma function:

$$\Gamma(z)=\int_0^\infty x^{z-1}e^{-x}dx.$$ It is a function defined in $\mathbb C$ by an integral (except for the negative integers), that when evaluated to any integer $n$, gives $(n-1)!$. We say it is actually the analytic continuation of the factorial function. Also, it verifies

$\Gamma(z)=z\Gamma(z-1).$

In fact, $\Gamma(4.5)\approx 11.6$ (note that we have to add 1 in the input to compare to the factorial).

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