How do we calculate factorials for numbers with decimal places?

Question

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?

Answer 1

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.