While moving my laptop the other day, I ended up mashing the keyboard a little, and by pure chance managed to do a google search for

`i!`

.Curiously, Google’s calculator dutifully informed me that i! was, in fact, 0.498015668−0.154949828i.

Why is this?

I know what a factorial is, so what does it actually mean to take the factorial of a complex number? Also, are those parts of the complex answer rational or irrational? Do complex factorials give rise to any interesting geometric shapes/curves on the complex plane?

**Answer**

It is sort of an abuse of what is meant by *factorial*. The usual definition of

n!=n∏k=1k

obviously cannot apply because you can sit and count integers until the end of time and beyond and you’ll never find i.

However, we can generalise what we mean by *factorial* by using a property of the gamma function, which is defined to be

Γ(z)=∫∞0e−ttz−1dt

This has the useful property that, for any n∈N,

Γ(n)=(n−1)!

which has an easy proof by induction on n. It also has lots of nice analytical properties which make it a good choice for an extension of the factorial function.

Anyway, since the gamma function can be defined (after analytic continuation; see LVK’s comment) on the entire complex plane, minus the non-positive integers, for a general z∈C−{−1,−2,⋯} we can put

z!def=Γ(z+1)

For this reason we get

i!=Γ(i+1)=∫∞0e−ttidt≈0.498015668−0.154949828i

**Attribution***Source : Link , Question Author : growse , Answer Author : Community*