/sci/ - Science & Math

asked cleo on stack exchange

>>8727634
I believe it relates to the Gamma function.

something with n! = n*(n-1)!
at least ??

>>8727634

Because once upon a time people knew about factorials, where you take whole numbers and multiply them together in certain ways. You have to have a whole number to start with, and what it means is, you take the first n whole numbers, and multiply them all together, and that's what you get. for example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

So obviously, just starting from here, taking the "factorial" of a number that isn't a whole number doesn't seem to make any sense. But if you plot the factorial points on a graph for a little bit, dot ... dot ... dot ... it is natural to wonder whether some /function/ (or, specifically, its graph) happens to /go through all of those points/, and preferably a /nicely defined thing/ of some kind.

Some big-shot math guys worked on figuring this out in the 18th century, and then eventually the big-shot math guy Euler came up with a calculus formula which works for pretty much all real numbers, and includes the "factorial" situation as a special case! It came to be called the /gamma function/, which is the extension of the idea of factorials, to other sorts of numbers (but which of course has to abandon the definition of the factorial in these cases). Nevertheless, the two ideas are so closely related that one might see the (slightly abusive) notation seen in the OP's picture.

>>8727634
Use the [math] \Gamma [/math] function.

>>8727639
This guy knows what's up

>>8727665
lol

>>8727654
Not OP. This is a really thoughtful answer. Thanks!

>>8727654

thanks

>>8727634
Well, once you know that:

[math]\frac{d^n}{dx^n} x^n = n![/math]

then you just plug in n=1/2


[math]\frac{d^{1/2}}{dx^{1/2}} x^{1/2} = (1/2)![/math]

>>8727692
thanks I get it

Why do math threads on /sci/ get helpful and informative answers but science/physics/medicine threads get shitposts and bullshit?

8 days remaining.

I am quite pleased with this branch of mathematics being discussed on this board finally.

>>8727705
Mathematics becomes far more intuitive than other sciences and is easier to come to a definitive conclusion. When speculation/vagueness is present, arguments arise.

>>8727692
half a derivative? Da fuq am I looking at

>>8727634
There's several ways I know how to derive it. They all involve doing integrals and using the definition that n! = L(n+1) where L is the gamma function. Do you want me to write any out for you?

>>8727740
the only way i know of showing this is turning it into the Gaussian integral, and doing a double integration to show that it equates to sqrt(pi). if you know another method, i'd love to see it.

t. not op

>>8727705
/pol/tards are too dumb to understand math so they don't post in these threads

>>8727634
There's a series of articles called something along the lines "How Euler Did It". One of these articles focuses on the derivation of the gamma function.
Basically, the original problem back in the day was to find the smooth, exponential looking function that connects all the points of the factorial function (which what your image is showing, as the factorial function itself is only defined for the integers). If I recall correctly, the approach Euler use was basically an interpolation procedure...it's been years though since I actually looked into the derivation of this function.

>>8727736
https://en.wikipedia.org/wiki/Fractional_calculus

>>8727719
>Mathematics becomes far more intuitive than other sciences
1+2+3+4+...=-1/12

>>8727883
I have wondered if people wanted me to weigh in on this but I tend to only answer questions that have been directly posed.

I assume you are presenting this as a counter-argument for mathematical intuition.

>>8727744
Sorry for the late reply. Here's another method:

Take Euler's integral B(m,n)=B(n,m)=integral x^(m-1)*(1-x)^(n-1) from 0 to 1
Substituting x/(1-x) or (1-x)/x will give
B(m,n)=integral x^(m-1)/(1+x)^(m+n) from 0 to infinity.

Now we'll come back to that later. It can be shown that B(m,n)=L(m)L(n)/L(m+n) where L is the gamma function.

Now, for the real beauty.
If n is even
Integral x^(m-1)/(x^n+1)=-1/m*{sum cos(mrB)*log(x^2-2xcos(rB)+1)} + 2/n*{sum sin(nrB)*arctan((x-cos(rB))/sin(rB)) where the sum's are done with respect to r for values r=1,3,5,...,-1 successively
B=pi/n

If we substitute m=2p+1 and n=2q and integrate this between + and - infinity the first term becomes log(1)=0 and the second becomes the series

pi/q{sin(n*pi/(2q)) + sin(3n*pi/(2q)) + ... + sin((2q-1)n*pi/(2q))} = pi/(q*sin(K*pi))
where K=(2p+1)/(2q)

Okay, now we must divide it by 2 because we integrated from - infinity to + infinity, but we only want from 0 to infinity.
Hence we have the result

Integral x^(2p)/(1+x^2q) from 0 to infinity equals pi/(2q*sin(Kpi))

Setting m+n = 1 in the gamma function yields L(1)*B(m,1-m)=L(m)*L(1-m)
L(2)=1 by definition.

Combining everything we have and putting m=1/2

B(1/2)=integral 0 to infinity x^(-1/2)/(1+x) = pi/(sin(pi/2))=L(1/2)*L(1/2)

From which we get L(1/2)=sqrt(pi)
EASY AS PIE

>>8727950
the sum r is done from 1,3,5,...,n-1

>>8727950
r goes from 1,3,5,...,n-1

When I said L(2)=1 I meant L(1)=1

By B(1/2) I meant B(1/2,1/2)

how do you guys use latex in 4chan?

>>8727961
Write whatever you want down in Wolfram Mathematica
Right click, copy as latex, paste here

>>8727968
Also, get the "4chanX" extension and click the button in the pic to preview what you're about to post

And finally, here's your free Mathematica 11 copy for taking the time to write such a long and helpful post
https://thepiratebay.org/torrent/15920734/Wolfram_Mathematica_11.0.1___Keymaker_[SadeemPC]
Might be temporarily down for maintenance, will come back up within 20-30 minutes

>>8727902
it was a meme but a lot of maths isn't intuitive, the -1/12 is a testament to this

>>8727968
thanks, got it

>>8727983
The -1/12 thing is only posted by people who don't understand where it comes from and what it means
Same with e^(i*pi)=-1omg so deep such a nerd XD

>>8728005
It's still incredibly counter-intuitive regardless of how you discover it

Gamma(n-1) = n!

>>8728066
>kek
Brainlet

Did someone misquote wikipedia?

>>8728005
Posts like these are only posted by people who don't understand what [math]e^{i \pi}[/math] or [math]- \frac 1 {12}[/math] actually mean to mathematics.

>>8727639
is her medical condition real?

>>8727950
very neat, i like that you didn't have to use double integrals

could i bother you to write that up in latex? use [ math] [/ math] tags to surround your eqn, pic related
[math] x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/math]

>>8728796
forgot pic bc idiot

It relates to the Gamma function through some Fourier series

>>8727634
Aside from using integration by parts with a trigonometric substitution on the gamma function, you might want to understand that permuting one half is abstractly similar to a demi-rotation in the plane. There is a hint as to why we can use the gamma distribution to model samples from an infinite set with a range of permutations.

All functions are smooth. Therefore multiplication is not valid. I win math.

>>8727654
why is the gamma function shifted by 1

>>8729153
this triggers my autism so bad

>>8729153

>>8728605
hmm

>>8727654

reminds me of

[math]e^{\pi i} = -1[/math]

It basically only works because they made up a definition for imaginary and real exponents

>>8728605
I have always wondered
All this beautiful mathematical equations that people pander only a "genius" would see
As for anything that can be found in life, its just a matter of making abstract combinations until something useful pops up, I am 99% sure that any impressive discovery we have ever made was just a "huh that's funny" coincidence accidental discovery
How close we are to make an algorithm that finds in 3 seconds all those important mathematical formulas we have made in all human history
You just have to make an AI that will think only in the most simple way that an idea can ever be represented, all people say that the most simple is bits, on-off and then grouping them next to each other, but there exist a simpler, first you start with "nothing" that's the simplest abstract idea that can be made, the you add "something" these 2 can be grouped to form one "something" meaning that at the same time we have created a "nothing"next to it, so now we have 2lvls, then we can continue grouping them to infinity(pic related), then with this you will be able to represent any data and create another form of maths that has as a building block a better simpler way to represent information

>>8727749
The only one bringing up /pol/ in here is you.

Why do you have to be so rude and disrespectful?

>>8729564
> triggered /pol/tard

>>8729571
That doesn't answer my question.

literally pic related

>>8729591
trivial

>>8727634
>.5!
That's undefined, factorials are only defined for the natural numbers and zero.
The gamma function is an extension of factorials to the real line, that has that value at a half.

>>8729685
And it should be said that there are several functions f(n) that interpolate n! nicely.
The gamma can be characterized via pic related, and it's also the function extending the polynomial theorem
(1+a)^s = \sum...
in the right way.

The fact that (1/2)!^2 is pi is probably Mochizui'esque shit, but you can trace how you get there
n! is the n's derivative (∂/∂x)^n of x^n and e^x is important in all things ∂/∂x. At the same time, exp generates motion on Lie groups like rotations and that's where things come together with pi.