Why is i!=0.498015668−0.154949828ii! = 0.498015668 – 0.154949828i?

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, 0.4980156680.154949828i.

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!=nk=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)=0ettz1dt
This has the useful property that, for any nN,
Γ(n)=(n1)!
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 zC{1,2,} we can put
z!def=Γ(z+1)
For this reason we get
i!=Γ(i+1)=0ettidt0.4980156680.154949828i

See also here and here.

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

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