TITLE: Why is $i! = 0.498015668 - 0.154949828i$?
QUESTION [76 upvotes]: 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?

REPLY [8 votes]: 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.