what is the value of i factorial using the complex number system?

Question

what is the value of i factorial? "I" belongs to the complex number system. Thanks for helping me out with this problem.

Answer 1

The Gamma function gives $$ \Gamma (1+i)=i\Gamma (i)\approx 0.498-0.155i $$ However, only for integers we have $\Gamma(n+1)=n!$. Nevertheless one may view this as a generalization of factorial. The question has already been answered in great detail in this sense here. There is also a duplicate at MSE:

Factorial of $i$

Answer 2

In mathematics, the factorial of "$\text{a non-negative integer}$" $n$, denoted by $n!$, is the product of all positive integers less than or equal to $n$.

That is, $n\in \textbf Z$ only. It is not defined for complex numbers. Complex numbers are out of the factorial function's domain.

I hope this helped you. Cheers!!