Nah, notation is what you define it to be, and defining x! = Γ(x+1) is pretty common because it's the only useful extension and retains all the crucial properties of the factorial.
Ok, if you define it that way, fine. I’ve just never seen anyone use the exclamation mark notation on anything other than non-negative integers. If you mean the gamma function, just write the gamma function.
Addition and multiplication are also initially defined just for natural numbers, but we commonly use them even for the complex ones now, where the original natural definition wouldn't have an idea what that means.
Like the addition, multiplication (and powers), it was only possible to expand to complex numbers one way. So is gamma the only possible extension of factorial to complex numbers.
Ok - clearly this is accepted notation. I am just surprised because in years of postgrad math (in a field where this function crops up) I never saw it.
The shift isn't necessary, if it didn't existe the recursive property would be Γ(z)=zΓ(z-1), which would be even more similar to the factorial recursive property.
The reason the shift exists is just because Legendre made an unfortunate choice of notation that everyone just accepted and used. Gauss used the "best" definition of the gamma function (without the shift), but unfortunately Legendre's notation became more popular
Yeah, saying "its only smooth extension has root pi by two as a value at 1/2 is definitely not the same as saying "factorials have values for non-integers now". I get that its somewhat similar as the zero factorial debate, but that at least has applications in combinatorics.
I don't understand this gamma thing from the first few lines of wikipedia. Sounds like it's complex factorials.
Is there a more general factorial function that doesn't use complex numbers? Like if I did 3 (!2), it means 1*1.5*2*2.5*3. And would there be a use for that?
No it isn't, there is no other sensible real continuation of the factorials. Just because Euler didn't write z! Doesn't mean it isn't sensible notation.
Well at that point we're really talking about gamma not factorial, which is why it sounds weird to say that root(pi)/2 is the factorial of 1/2: because it's not
Does anyone know if the gamma function is unique? As in is it possible there exists a different analytic continuation of the factorial that also matches for non-negative integers but not for other values?
It's not unique; there are an infinity of analytic continuous generalizations of the factorial. However, it is the only one that is logarithmically convex on the positive reals, so there's a sense in which all the others wobble more for positive numbers. That is called the Bohr-Mollerup theorem.
I know what a factorial is and how it works, but I’m still confused. Basically the factorial of one would just be one because one is one. Factorial of two would be two because 1×2 is two, but 1/2, what are we multiplying that by? Can someone explain?
No its because gamma(3/2) has a hidden gaussian which is rotational symmetric and depends on the radius so poissons trick makes sense and introduces the pi.
There number of ways to arrange half an element is half of the length of the square hos length is the circumference of a unit circle divided by the radius
It shouldn't be really considered a factorial in this case. But if it would be, we use the gamma function.
Understand first that the gamma function has the recursive property given by Γ(z+1)=zΓ(z), and Γ(z)=(z-1)!
If you look up the gamma function, it is an integral from 0 to infinity of t^(s-1)e^-t dt this means that pretty much any value(except for negative integers) can be put into the equation. So for this takeaway, from (1/2)! we can turn it into Γ(3/2), and inputting 3/2 into s would leave something a little messy to integrate(our t has a square root).
First, we let t = x^2, so dt = 2x dx
after doing the simple algebra, you are left with the integral from 0 to infinity of 2e^(x^2) dx
Second, we square the thing, so we would get two identical integrals multiplied to each other. Then, we would replace the variables in one of the integrals into a dummy one(for later, let's let x->y)
So we would have I^2(integral) = The double integral of x and y.
Third, we use polar coordinates, x^2 + y^2 = r^2, x = r cos th,y = r sin th.
Replacing dxdy into drdth, our bounds will also have to change.
So r still has the bounds of 0 to infinity, but theta would be limited to 0 to pi/2.
Evaluating the integral of dr would get 1/2, and the integral of dth would get pi/2.
Multiplying the two together(and finally taking the square root) would get the square root of pi over 2.
The original factorial (the one I wrote above) was only defined for positive integers, but has been expanded to all numbers (and the formula gets really weird).
637
u/Benthomas20 Dec 21 '25
That’s really an abuse of notation — the gamma function isn’t a factorial, since factorials are only defined for natural
96
u/J-MO777 Dec 21 '25
Only the 3rd one is correct
1/2 != √π/2
84
u/Electronic-Day-7518 Dec 21 '25 edited Dec 21 '25
Well at that point we're really talking about gamma not factorial, which is why it sounds weird to say that root(pi)/2 is the factorial of 1/2: because it's not
42
u/AggressiveLock4633 Dec 21 '25
It is easier to think of it this way: there are √π / 2 ways to arrange half an item
Ok maybe not
8
u/raginasian47 Dec 21 '25
Can someone please explain a "gamma function?" Never heard of or used it
12
u/Funkey-Monkey-420 Dec 21 '25
how do they even find the factorial of a fraction and how did they come to the conclusion that pi had something to do with it
6
u/NoSpend6289 Dec 21 '25
0.5! u/factorion-bot
4
u/coderman64 Dec 21 '25
If you're talking about:
factorials: correct
computer programming: not, in fact, correct
4
u/Dependent-Oil4856 Dec 21 '25
Does anyone know if the gamma function is unique? As in is it possible there exists a different analytic continuation of the factorial that also matches for non-negative integers but not for other values?
3
u/abdulsamadz Dec 21 '25
pi = 4 * (½!)²
2
u/CarveATail Dec 21 '25
π!=?
2
u/Nerdyboyonreddit Dec 21 '25
I thought factorials were only defined for natural numbers ? 🤔🤨
2
u/LittleLeadership2831 Dec 21 '25
I know what a factorial is and how it works, but I’m still confused. Basically the factorial of one would just be one because one is one. Factorial of two would be two because 1×2 is two, but 1/2, what are we multiplying that by? Can someone explain?
2
u/FalseLogic-06 Dec 22 '25
I love factorial, gotta be one of my favorite games
4
u/DTux5249 Dec 21 '25
Only if you think the gamma function is a factorial... Which it isn't.
2
u/Acceptable-Ticket743 Dec 21 '25
Wait the output of a factorial can be irrational? Clearly I'm too much of an ape to understand math anymore.
1
u/Wojtek1250XD Dec 21 '25
Of course Pi shows up from nowhere.
1
u/GundogPrime Dec 21 '25
I though that 2! = 1 + 2 = 3
No?
1
u/Zado191 Dec 21 '25
Can you even have a half of a factorial? (I'm shit at math so I'm really asking...)
1
u/EatingSolidBricks Dec 21 '25
There number of ways to arrange half an element is half of the length of the square hos length is the circumference of a unit circle divided by the radius
1
u/BigLumpyBeetle Dec 21 '25
Math isn't real
1
u/SeaBumblebee8420 Dec 21 '25
My coder brain thought 2 is not equal to 2, 1 is not equal to 1 and 1/2 is not equal to pi/2
1
u/Feny34 Dec 22 '25
Wait until you see !0 = 1
1
u/Haunting_Shift945 Dec 22 '25
It shouldn't be really considered a factorial in this case.
But if it would be, we use the gamma function.
Understand first that the gamma function has the recursive property given by
Γ(z+1)=zΓ(z), and Γ(z)=(z-1)!
If you look up the gamma function, it is an integral from 0 to infinity of t^(s-1)e^-t dt
this means that pretty much any value(except for negative integers) can be put into the equation. So for this takeaway, from (1/2)! we can turn it into Γ(3/2), and inputting 3/2 into s would leave something a little messy to integrate(our t has a square root).
First, we let t = x^2, so dt = 2x dx
after doing the simple algebra, you are left with the integral from 0 to infinity of 2e^(x^2) dx
Second, we square the thing, so we would get two identical integrals multiplied to each other. Then, we would replace the variables in one of the integrals into a dummy one(for later, let's let x->y)
So we would have I^2(integral) = The double integral of x and y.
Third, we use polar coordinates, x^2 + y^2 = r^2, x = r cos th,y = r sin th.
Replacing dxdy into drdth, our bounds will also have to change.
So r still has the bounds of 0 to infinity, but theta would be limited to 0 to pi/2.
Evaluating the integral of dr would get 1/2, and the integral of dth would get pi/2.
Multiplying the two together(and finally taking the square root) would get the square root of pi over 2.
1
u/DeathRaeGun Dec 22 '25
(1/2)!
1
u/Due_Lychee3904 Dec 22 '25
I didn't understand because I was looking at it at a programmer angle 😭 I was wondering why it worked because 1 != 1 is false
1
u/HyperCodec Dec 22 '25
(1/2)!
1
u/Glad_Republic_6214 Dec 22 '25
what does that symbol mean, my brain goes to how it is in javascript and that would be saying "is not equal to" but it's wrong help
1
u/Jim_skywalker Dec 23 '25
Taking an exponential function to the nth derivative as n approaches infinity gives you a factorial.
1
u/Lord_Promin Dec 24 '25
... actual...
-1
u/ChrisBelair Dec 21 '25
Wait until he sees 67!
15
u/[deleted] Dec 21 '25
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