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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    $\begingroup$ Google "Gamma Function" $\endgroup$ – Pedro Tamaroff Sep 25 '12 at 12:39
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    $\begingroup$ This should give you a jumping off point: wolframalpha.com/input/?i=i%21 $\endgroup$ – Austin Mohr Sep 25 '12 at 12:40
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    $\begingroup$ Strictly speaking, under the usual definition the factorial is only defined for natural arguments, so you will have to use a generalized definition for $i!$. $\endgroup$ – akkkk Sep 25 '12 at 12:43
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    $\begingroup$ $\Gamma(i+1)$ of course. $\endgroup$ – i. m. soloveichik Sep 25 '12 at 12:44
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    $\begingroup$ I changed the "number theory" tag to "complex analysis". $\endgroup$ – Michael Hardy Sep 25 '12 at 15:14

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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    $\begingroup$ The result comes just from numerical computation? Can it be expressed in terms of elementary functions or well known constants? $\endgroup$ – leonbloy Sep 25 '12 at 13:41
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    $\begingroup$ @leonbloy: Yes it's computed numerically. If there is a nice expression for it then neither I nor Google nor Wolfram Alpha knows what it is ;) $\endgroup$ – Clive Newstead Sep 25 '12 at 13:44
  • $\begingroup$ This is interesting stuff, but I have a follow-up: is factorial the only case where a function which only works on a certain set of numbers has been 'abused' and turned into a different function which also works across a bigger set? Or is factorial a bit unique here? $\endgroup$ – growse Sep 25 '12 at 15:19
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    $\begingroup$ @growse Riemann zeta-function is another example where original definition works only in some range of numbers, but can be profitably extended. "Abused" really means "extended" here. $\endgroup$ – user31373 Sep 25 '12 at 15:21
  • $\begingroup$ No need to exclude $0$ from $z!=\Gamma(z+1)$, by the way. $\endgroup$ – user31373 Sep 25 '12 at 15:21

$$i!=\Gamma(i+1)=\int_0^{\infty}e^{-x} x^{i}dx$$ 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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    $\begingroup$ How is that helpful? $\endgroup$ – Pedro Tamaroff Sep 25 '12 at 13:07
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    $\begingroup$ @PeterTamaroff: The OP asked, "what does it actually mean to take the factorial of a complex number?" And this answer helpfully but tersely says that one way to extend factorial is using the gamma function, and tells what the gamma function is. $\endgroup$ – LarsH Sep 25 '12 at 15:53
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    $\begingroup$ @LarsH Are you sure? It only see an equation with an integral, but no explanation of what that really means. It is not even mentioned that $\Gamma$ is the Gamma function, for example. $\endgroup$ – Pedro Tamaroff Sep 25 '12 at 16:11
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    $\begingroup$ @PeterTamaroff: "only an equation with an integral"? Are you implying that equations are not helpful?? The answer (even before the edit) showed the Greek letter gamma with function notation, which is information answering the OP's question. You asked "How is that helpful?", not "How is that maximally helpful?" Arguably, the equation defining the gamma function is just as helpful as the English phrase "Gamma function", if not more so. The latter is useful in searching; the former explains what function Google was evaluating. $\endgroup$ – LarsH Sep 25 '12 at 17:57
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    $\begingroup$ @LarsH I'm implying that if someone asks "How/why is $i!=\text{bleh}$"? then it is probably because they know nothing about the Gamma function and thus answering $$i! = \Gamma \left( {i + 1} \right) = \int\limits_0^\infty {{x^i}{e^{ - x}}dx} $$ is pretty useless. Let alone that the OP is not supposed to know $\Gamma$ is the Greek letter gamma. Lately we seem to think leaving short "smart" answers that a few can understand (and will probably upvote) will help the OP, and that is not the case. Compare to Clive N.'s answer. $\endgroup$ – Pedro Tamaroff Sep 25 '12 at 18:03

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.


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