Does this need to do anything v the gamma function? there are specifically sqrt(Pi)/2 methods to arrange 1/2 objects... Yeah, ns can't comprehend.

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The means to extend a role from a smaller sized domain come a bigger one is to discover some kind of formula that has the role in the that, in part way, suggests what the duty should be at values outside of it's initial domain.

Let's say that we desire to know what x-n needs to be, assuming we only know that xn=x multiply by chin n times and also that x0=1. We recognize that the xa+b=xaxb. This says that xnx-n=x0=1. So we have actually the formula xnx-n=1 and the just thing that is not characterized in this is x-n. Therefore we deserve to solve because that it and arrive in ~ the formula x-n=1/xn and also we have effectively defined an adverse powers in a natural method by making use of an extension formula. This happens all the time in math.

So if we want to expand the factorial role to include positive half-integers, then we deserve to do so there is no imposing any kind of Gamma function magic. This is excellent by relating the volume that n-dimensional ball to factorials. Let's have V(n)=Volume of the n-dimensional ball of radius 1. For this reason V(1)=2 (this is simply the term <-1,1>) and V(2)=pi, V(3)=4pi/3 etc. Deserve to we find a basic formula because that this in state of simply n?

It turns out that we can display that

V(2k) = pik/k!

V(2k+1) = 2(k!)(4pi)k/(2k+1)!

The formula because that the even-dimension situation is really nice, yet the measurement for the strange dimensional situation is really ugly, and also the truth that lock are split like this kinda sucks. Is over there a much better way that we have the right to write this that doesn't distinguish between these instances so the I deserve to just compose "V(n)=..."?

If n is even, then I can rewrite the even instance as V(n)=pin/2/(n/2)! and this is okay since n/2 is still an integer. This formula is nice. I desire it to job-related for every cases. Due to the fact that (n/2)! is not identified when n is odd, I'm totally free to assign to it every little thing number i want. So I'm walk to specify (n/2)! to be everything number it needs to it is in so that V(n)=pin/2/(n/2)! for any n.

This method that when n is odd, i need

2((n-1)/2)!(4pi)/2/n! = pin/2/(n/2)!

The truth that n is odd way that every little thing on the left side of this equation is defined. In fact, the just thing that is not defined in this equation is (n/2)!, for this reason we have the right to solve because that it to view what it has to equal. Act this gives

(n/2)! = sqrt(pi)n!/(2n((n-1)/2)!)

This formula because that half-integer factorials is make so that V(n)=pin/2/(n/2)! for all integers n. Plugging in assorted odd n gives

(1/2)! = sqrt(pi)/2

(3/2)! = 3sqrt(pi)/4

(5/2)! = 15sqrt(pi)/8

Obviously, the worths of (n/2)! execute not phone call us just how sets of dimension n/2 can be ordered, since no together sets exist. But the figure of factorials in the formulas for the volume are associated to orderings of sets, so we're essentially finding a formula for how plenty of ways a set of dimension n/2 would be ordered if it did exist. Generally, whenever you check out a factorial, you're counting permutations of set somehow, for circumstances Here's why they space in Taylor Series. Similarly, whenever you watch pi in a formula climate you're law something with circles. V this method, it's clean why there has to be a pi in the formula for (n/2)!, since we're using the recipe for volumes of round to do it!

This technique of prolonging the factorials is obviously restricted to just half-integer factorials. But it's more naturally motivated than simply pasting the expression because that the Gamma duty and just saying "Plug in x=3/2" and it tells us how pi normally gets involved. The resulting formula is something that is motivated, and also not just handed down from Euler, our Lord and also Savior.

But us can usage the Gamma function to extend every one of this also further, it offers us a factorial for every facility number other than the an adverse integers. In fact, over there is a theorem saying that the Gamma role is the only means to prolong the factorial to the complex plane in a "nice enough way". But many of these type of results, worths of familiar functions at unfamiliar values having unintuitive values, deserve to be done without appealing to modern-day tools. Our modern tools space super an effective and were emerged to find answers to very hard questions, like the element Number theorem or the Riemann Hypothesis. Lock weren't created to advice the factorial role at non-integer values, they simply need to be continual with other, easier methods that execute this, but they sell no cool math-tricks or insights. Simply plug-and-chug.

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I additionally find it weird that human being are okay through the formula (1/2)!=sqrt(pi)/2 and not okay through the formula 1+2+3+4+...=-1/12. Come me there's no philosophical difference, both are extending previous notions to where they room undefined by detect some expansion relation, yet people get antagonistic around 1+2+3+4+.. And also not (1/2)!.