I obtained a bizarre graph of $x!$, a function which I believe is only defined at positive integer domain.
What causes such an error?. It would be interesting if someone can explain the method such a graphing software uses to compute $x!$
I have attached the image of the graph below.
2 Answers
$\begingroup$You are right that, by the traditional definition of $x!$, it is only defined at the integers. However, we can analytically continue the factorial function to all real numbers (except, as Peter notes in the comments, for all negative integers and $0$). This continuation (along with a shift) is known as the Gamma Function.
However, note that there are other continuations of the factorial function. The one pictured here is the Gamma Function. Here are some other continuations which interlope $x!$.
$\endgroup$ 10 $\begingroup$I guess that program is Desmos.
The graph is alright. It's because of the interesting relation between Gamma function and the factorial function. Gamma of x is denoted by $\Gamma{(x)}$ The relation is $\Gamma{(x+1)}=x!$. This is often used to extend the domain of the factorial function. Using this we can prove that $(\frac{1}{2})!=\frac{\sqrt{π}}{2}$
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