I've been told that factorials of negative numbers doesn't exists that's what I also found while trying to calculate factorial of negative $1$. But, I can see that graph of factorial $x$ is even extended to negative side of $x$ axis. What's the reason for this?
But, Desmos is able to calculate factorial of floating point values such as $0.1 , 0.2 , 0.3, 0.4$ etc.
I'm aware of gamma function but not well verse will integrals.
Can anyone explain this?
Best Answer
The "usual" definition of factorial which one is first exposed to only works for positive integers. That is, if we construct the factorial as $n!=1\cdot2\cdot3\dotsm (n-1)\cdot n$ then obviously you cannot extend this to all real numbers. However, we observe that $(n+1)! = (n+1) n!$, so we construct a function $f$ defined for all real numbers* such that $f(x+1)=(x+1) f(x)$ and $f(1)=1$, by analogy with the recurrence for the factorial. This is essentially the Gamma function, and in fact works for all* complex number arguments, not just real numbers. It is useful in a variety of applications.
So, this definition of the factorial of any complex number is not the same as the usual notion of factorial: in particular it does not have a combinatorial interpretation of "all ways to permute $n$ distinct objects". But it is a generalisation which people have found to be useful.
*Technically, there is a small set of values for which this function cannot be defined (or has value infinity). These are the vertical asymptotes you see in the graph, and are called "poles".