Why is $i! = 0.498015668 - 0.154949828i$?

Question

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 1

It is sort of an abuse of what is meant by factorial. The usual definition of $$n! = \prod_{k=1}^n k$$ 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 $$\Gamma(z) = \int_0^{\infty} e^{-t}t^{z-1}\, dt$$ This has the useful property that, for any $n \in \mathbb{N}$, $$\Gamma(n) = (n-1)!$$ 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 $z \in \mathbb{C} - \{ -1, -2, \cdots \}$ we can put $$z! \overset{\text{def}}{=} \Gamma(z+1)$$ For this reason we get $$i! = \Gamma(i+1) = \int_0^{\infty} e^{-t}t^{i}\, dt \approx 0.498015668−0.154949828i$$

See also here and here.

Answer 2

$$i!=\Gamma(i+1)=\int_0^{\infty}e^{-x} x^{i}dx$$ where $\Gamma(n) $ represents the Gamma Function

Note $$x^i=e^{i\ln x}=\cos(\ln x)+i\sin(\ln x)$$

Answer 3

To answer your last question,

Do complex factorials give rise to any interesting geometric shapes/curves on the complex plane?

There are a couple of Gamma fractals shown on Wolfram's reference article for Gamma under "Neat Examples":

• DensityPlot[ Arg[Nest[Gamma, x + I y, 3]], {x, -1.25, -0.6}, {y, -0.25, 0.25}] // Quiet
• ArrayPlot[ Table[c = N[cr + I ci]; Length @NestWhileList[ If[Abs[#] > 20., Indeterminate, Gamma[#/c]] &, c, (# =!= Indeterminate) &, 1, 20], {ci, -2.5, 2.5, 5/100}, {cr, -2, 2, 4/100}]] // Quiet

See also Christopher Olah's blog post, Gamma Fractals, and from there, Bidimensional zoom in on the Z=Gamma(Z) iteration with display of the arguments, both of which have some nice images.