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?
1 Answer
$\begingroup$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".
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