What is (-π)! demonstration?

Question

(-π)!

Maybe

  (-π)! = (-3.1415926535897932384626433832795)! = -3.1909066873083528993320905932927

if this result is correct then

 What is the demonstration?

Answer 1

It's nonsensical.

But often, factorial notation is used to mean the gamma function rather than factorials: specifically, $n!$ is used to mean $\Gamma(n+1)$

Answer 2

The factorial function is generally understood to be defined only for the domain $\{0,1,2,\ldots\}$. However, it coincides with another function, the gamma function, which is defined for all real numbers except for non-positive integers. In particular, $n!=\Gamma(n+1)$. Thus, we could identify $(-\pi)!$ as $\Gamma(-\pi+1)$.

The value of this function would be the number you wrote down.

Answer 3

The factorial is usually considered to be defined only for natural numbers. One standard way to extend the definition of factorial outside this set is to consider the gamma function: $\Gamma (x+1) = x! $ when $x $ is a natural numbers but is also defined, for example, when $x=-\pi $. In fact, $\Gamma(-\pi+1)\simeq -3.19$. With a slight abuse of notation whoever our whatever told you that $(-\pi)!\simeq -3.19$ most likely adopted the extended definition provided by the gamma function.