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From what I know, the factorial function is defined as follows:

$$n! = n(n-1)(n-2) \cdots(3)(2)(1)$$

And $0! = 1$. However, this page seems to be saying that you can take the factorial of a fraction, like, for instance, $\frac{1}{2}!$, which they claim is equal to $\frac{1}{2}\sqrt\pi$ due to something called the gamma function. Moreover, they start getting the factorial of negative numbers, like $-\frac{1}{2}! = \sqrt{\pi}$

How is this possible? What is the definition of the factorial of a fraction? What about negative numbers?

I tried researching it on Wikipedia and such, but there doesn't seem to be a clear-cut answer.

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the definition is the definition of the gamma function. –  Integral May 20 '13 at 3:12
Gamma functions is sort of an extension of the concept of factorials to fractions. –  Uma kant May 20 '13 at 3:12

2 Answers 2

up vote 19 down vote accepted

The gamma function is defined by the following integral, which converges for real $s>0$: $$\Gamma(s)=\int_0^\infty t^{s-1}e^{-t}dt.$$

The function can also be extended into the complex plane, if you're familiar with that subject. I'll assume not and just let $s$ be real.

This function is like the factorial in the when $s$ is a positive integer, say $s=n$, it satisfies $\Gamma(n)=(n-1)!$. It generalizes the factorial in the sense that it is the factorial for positive integer arguments, and is also well-defined for positive rational (and even real) numbers. This is what it means to take a "rational factorial," but I would hesitate to call it that. Many functions have those two properties, and $\Gamma$ is chosen out of all of them because it is the most useful in other applications. Rather than the notation used in that article you refer to, it would be more accurate for you to say that "the gamma function takes these values for these arguments." Gamma is not a function that intends to generalize factorials; rather, generalizing factorials came along as something of an accident following the definition. Its true purpose is deeper.

As for why $\Gamma(1/2)=\sqrt{\pi}$, this comes out of an interesting property of the $\Gamma$ function: some of them are here http://en.wikipedia.org/wiki/Gamma_function#Properties. The property you are interested in is the reflection formula: $$\Gamma(1-z)\Gamma(z)=\frac{\pi}{\sin(\pi z)}.$$ Set $z=1/2$ in the formula to get the desired identity.

If you want to learn more about the gamma function, the hard way is to learn a lot more math, in particular real and complex analysis. An easier way is to read this excellent set of notes: http://www.sosmath.com/calculus/improper/gamma/gamma.html.

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Both ways are rewarding, the former perhaps exponentially more so. –  neuguy May 20 '13 at 3:27
It's interesting how our answers have exactly the same number of votes (6) at this point. I think I prefer yours. –  Lee Sleek May 20 '13 at 3:40
@LeeSleek Thank you. I like yours too, infinite products are beautiful things. –  neuguy May 20 '13 at 3:43
7 to 5, I'm guessing that was the same person upvoting and downvoting. (I voted for your answer.) –  Lee Sleek May 20 '13 at 3:46
@JpMcCarthy You'd get a better and more detailed response if you posted this as a new question. The best answer I can give you right now is that, like I've mentioned in my answer, $\Gamma$ was not defined to generalize factorials. If you're still not satisfied, you can define $\Delta(x) = \Gamma(x+1)$, and then $\Delta$ will satisfy $\Delta(n) = n!$. But I don't see that there's any value to $\Delta$ over $\Gamma$, especially since the analytic continuation of $\Gamma$ is aesthetically nicer (cuts off at real part $0$ nicely). –  neuguy Nov 13 '13 at 11:23

The gamma function, shown with a Greek capital gamma $\Gamma$, is a function that extends the factorial function to all real numbers, except to the negative integers and zero, for which it is not defined. $\Gamma(x)$ is related to the factorial in that it is equal to $(x-1)!$. The function is defined as

$$\Gamma(z) = \frac{1}{z} \prod_{n=1}^\infty \frac{\left(1+\frac{1}{n}\right)^z}{1+\frac{z}{n}}$$

Simply use this to compute factorials for any number. A handy way of calculating for real fractions with even denominators is:

$$\Gamma(\tfrac12 + n) = {(2n)! \over 4^n n!} \sqrt{\pi}$$

Where n is an integer. But keep in mind that the gamma function is actually the factorial of 1 less than the number than it evaluates, so if you want $\frac{3}{2}!$ use n = 2 instead of 1.

Or, you could just put the fraction into Google Calculator, which uses the gamma function to evaluate factorials of any number.

For some more examples of the gamma function's values, see here.

(If you don't understand this, don't worry, because I don't either, and the Wikipedia article on the function seems to lack a clear-cut definition of it or how it relates to $\sqrt{\pi}$.)

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'If you don't understand this, don't worry because I don't either'? Perhaps the questioner wants to know more about the topic than you do? –  jwg May 20 '13 at 8:19
It would be nice to know why the infinite product is equal to the factorial for natural numbers, and why $\Gamma(n) = (n-1)\Gamma(n-1)$ for all values. The formula you gave works for real fractions with denominator $2$, not any even number. If you can't say anything about how to evaluate infinite products, about where that product comes from, the poles of the Gamma function, the relationship with $\sin$ or the product expression for $\pi$, it's not clear why you bothered to try and answer. –  jwg May 20 '13 at 8:24
...And that's why I voted for proximal's answer as the best. –  Lee Sleek May 21 '13 at 19:08

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