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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?

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    (About that second approximate-equality: $1.00001^5\approx1.00005$, for example. Check this with a calculator.)2015-02-24

3 Answers 3

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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.

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    @CliveNewstead What numerical technique could one use to solve the above integral? Can one use Riemann Sums, or can Riemann sums only be used for real valued integrals?2016-01-23
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$i!=\Gamma(i+1)=\int_0^{\infty}e^{-x} x^{i}dx$ where $\Gamma(n) $ represents the Gamma Function

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

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    @LarsH Whatever floats your boat, man. We cannot agree on this one, it seems. I told the OP to google it for I was on my phone so it was utterly tedious to write down an answer, if you're concerned.2012-09-25
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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".

See also Christopher Olah's blog post, Gamma Fractals.