/sci/ - Science & Math

Make me

Why the FUCK did these faggots shift it by one.
Even if you look at the integral definition of gamma function it would have been easier to have Gamma(z)=z! but no they didn’t fucking do it

>>14709635
Pi function

I have a math degree and I've encountered the gamma function exactly twice (volume of n-spheres and gamma distribution). Is it an overrated meme or does it only play a more prominent role in obscure number theory no normal person cares about?

>>14709683
Ramanujan apparently mentioned he had used the gamma function to discover properties about numbers, which made Hardy scoff since the gamma function is widely known, but when he read the results he realized they were new

>>14709635
The Gamma function is the Mellin transform of the exponential, which is defined the way it is because of its scaling properties.
It's better to write the integral as
[eqn]\Gamma(s) = \int_0^{\infty} x^s e^{-s} \, \frac{dx}{x}[/eqn]
[math]\frac{dx}{x}[/math] is the invariant measure under scaling.

>>14709683
it is used in physics iirc

>>14709692
>which made Hardy scoff since the gamma function is widely known
academics try not to be anti intellectuals challenge (impossible)

>>14709683
literally everywhere in analytic number theory

>>14709617
That's a true statement, nothing to discuss

>>14710450
You are to discuss the relative merits of the reasonable z! notation and the bizarre shifted Gamma function notation.

>>14710456
That wasn't stated in OP. The game function is true and meritable. If you're not used to the +1, then why not just abandon the factorial notation and get used to actually using gamma like the rest of the world and stop living in only the integers

>>14710460
Gamma* ffs

>>14710460
I am the OP.

>>14710460
Also... I don't live "only in the integers". Point is to use the factorial notation for all complex z for which the Euler Gamma product is defined.

>>14710467
And you didn't say it in the Original Post (OP)
>>14710469
>use the factorial notation for all complex z
Okay so what's (sqrt(2)-6.735i)!? Do you seriously not see issues with using factorial notation? Have you never had a professor go through a problem and end up with a surprising 0 and note it with an exclamation mark making it 0! which is then 1 and false for the problem? Or similarly in text ending the sentence with a 0!? Fuck factorial notation

>>14710485
I do not see any issues with using the factorial notation any more that I see issues using the standard [math]w^z[/math] notation for complex exponents.

>>14710548
Okay well you didn't give me a value for what I asked in factorial notation and you just ignore the issues I brought up. What issues do you have with complex exponents? Or do you just not see issues with anything and shouldn't be having discussions about how to best represent mathematical notation?

>>14710580
It's roughly -0.00212 - 0.00136i. The issues you mentioned are made up nonsense that aren't actually issues. I mentioned complex exponents because the point is we use same notation for complex exponents as we do for real, rational and integral exponents.

>>14709683
It's all over statistics dude.

>>14710643
My point isn't the generalization of variables and use of same notation, my point is factorial notation sucks and just get used to the gamma function

>>14710643
How did you get that value with factorial?

>>14709617
>Write a boring looking math equation
>Leave out the line in the middle of the "F"
>Suddenly it looks all cool and spooky

>>14710682
>Never seen a Greek letter before
You must be 18 to post here

>>14710682
[math]\kappa \iota \ell \ell \ \Upsilon o \upsilon \mathbb{R} \zeta \varepsilon \ell \int [/math]

>>14710681
How do you compute the gamma function? Come on...

>>14709683
>I have a math degree and I've encountered the gamma function exactly twice
clown school

>>14710792

>>14710753
Often times numerically, but that's the point, the gamma function you can, factorial you cannot. You rely on the generalization and call the two the same then go to gamma to actually get the values and hate on the gamma everywhere

>>14711005
What the hell are you saying you utter moron. z! is defined much the same way as the gamma function. This discussion is about NOTATION.

>>14709715
This.
The melanin transform made gamma relevant.
People were using capital pi for the generalization of factorial since it seemed more natural to use s instead of s-1 in the integral.
I still like the capital pi notation because Pi(x)*Pi(-x) = 1/sinc(pi*x) and sinc(pi*x) just so happens to be the fourier transform of the unit box centered at 0 (which looks like a capital pi).

>>14711207
Where is factorial defined on non-integers in publication without defaulting to the gamma function?

>>14709617
You can just use z!, it’s not hard to shift everything back.

>>14709715
>dx/x

This is your brain on incurable terminal autism

>>14711214
LITERALLY USE ANY OF THE SAME FORMULAS USED TO DEFINE THE GAMMA FUNCTION. You are the thickest motherfucker.

>>14712476
Then why do you bitch so much about the gamma function when that's literally the only thing you actually know?

>>14712959
Take a look at the fucking OP image again. This has literally been a thread about NOTATION from post #1 onwards. Stop shitting up my thread faggot.

>>14712963
Yes, and I keep saying to just use gamma and you keep bitching about it