[Razor]

Ah crap. Math sucks. If I were like CP, and did all this math crap, I would have gone crazy 2 days later.

[Neon]

*picks up shotgun from under the seat and tries to shoot down this thread*

[Sandwich]

Guys, quit the math/CP bashing - now.   Like Venom said in his Anime render thread - if you don't like it, don't go there. Personally, while I don't understand a single thing here, I find it fascinating that people are applying their minds to something actually useful outside of render and 3D modelling techniques. I can understand if this was a thread about something else, and maths was brought into the discussion just out of spite, but it isn't, and it wasn't, so cut it out.

[CP5670]

Thanks, Sandwich, although I don't really mind it too much.

Some of us less enlightened (but no less interested in mathematics) haven't heard of some of these things. Links are A-1 SUPAR. For example, what is G, the gamma function, the beta function, and Khinchin's Law?

On that note, thanks for describing K(x). What is it used for?

Sure; I don't have much time at the moment, but I will point you to some links.

The gamma function is basically an extension of the usual factorial to noninteger arguments, and the two are related by G(x+1) = x! . There is a bunch of information about this one available here:
http://mathworld.wolfram.com/GammaFunction.html

There exist several formulas and identities for this function as well, some of which can be used to prove that product result I gave earlier.

G is called Glaisher's constant and is actually defined by that limit; I think it is not known whether the constant is irrational, (although it is likely to be). It can be written in terms of derivatives of the zeta function, but not in terms of any of the usual functions or constants. I would post a link for this as well but the site I usually go to for constant-related information is currently down.

Here's one neat identity I can remember at the moment:

(pi and e you know, G is Glaisher's constant, and g is Euler's constant)

As for Khinchin's Law, it has to do with continued fractions. A CF is an "infinite fraction" of the following form:


Just like there is a decimal representation for every number, there is also a CF representation. For rational numbers, the CF does not go on forever but terminates. It turns out that, while irrational numbers have no periodic repeats in their decimal form, such patterns do frequently exist in CF form, and many common roots and constants can be expressed as CFs with some pattern in them. What Khinchin found is that, for almost all real numbers, taking the geometric mean of the numbers in the CF (a₁, a₂, a₃, etc.) actually yields a special constant that is independent of the original number. In fact, if you select a real number completely randomly, the probability that it will satisfy Khinchin's law is actually 100%. (there are numbers that do not satisfy it, but there are infinitely more that do satisfy it) The constant is called Khinchin's constant and is given by the following infinite product:


It is not known to be expressible in terms of any other functions or constants, although there are a few really messy integrals and sums that give it.

I am not too experienced with this stuff myself so I do not know exactly why this law is true, but you can find some more information here: http://mathworld.wolfram.com/KhinchinsConstant.html


If you find this stuff interesting, let me know, as I can go on about this all day.

Are you sure you have the series correct?

looks like I wrote that one in all wrong; it should be ( e^p - e^-p ) / (2p), or sinh(p)/p. sorry about that; I think those errors got buried in all the html/vb tags.

[Joey_21]

Originally posted by CP5670
looks like I wrote that one in all wrong; it should be ( e^p - e^-p ) / (2p), or sinh(p)/p. sorry about that; I think those errors got buried in all the html/vb tags.

Ah, ok, that works.

I went to the 100000000th term in the series and came up with 3.1415926460901864... after applying the inverse hyperbolic sine to it.



How do you generate Gamma(x) in terms of a series and whatnot?

[Goober5000]

Yeah, this is cool stuff.

[Sandwich]

Originally posted by CP5670
G is called Glaisher's constant and is actually defined by that limit; I think it is not known whether the constant is irrational, (although it is likely to be). It can be written in terms of derivatives of the zeta function, but not in terms of any of the usual functions or constants. I would post a link for this as well but the site I usually go to for constant-related information is currently down.

Y'know, you should offer your services as a tech description writer - you certainly have a handle on all the techno-jargon.

[CP5670]

How do you generate Gamma(x) in terms of a series and whatnot?

For numerical computations, there is something called Stirling's asymptotic series that is valid for about z>1/2, and gets very accurate for higher values: (this is actually based on the limit I posted with the factorial; it converges on order x^-5)



To get negative values using that, use the following relation: (this one is an exact formula)

G(-z) = - p csc( pz ) / G(z+1)

Y'know, you should offer your services as a tech description writer - you certainly have a handle on all the techno-jargon.

lol thanks, maybe I should try writing bits of this stuff in the weapon descriptions...

[Joey_21]

Originally posted by CP5670


For numerical computations, there is something called Stirling's asymptotic series that is valid for about z>1/2, and gets very accurate for higher values: (this is actually based on the limit I posted with the factorial; it converges on order x^-5)



To get negative values using that, use the following relation: (this one is an exact formula)

G(-z) = - p csc( px ) / G(z+1)

Would you mind posting a link to where I could find this series?

[CP5670]

it looks like all the images on that site were switched around; I better edit my posts...

one moment...

[edit] okay, I updated the links; let's hope they stay like that now...

[Joey_21]

hmm how exactly do you get

(1+1/(12z)+1/(288z²)-139/(51840z³)-571/(2488320z⁴))

aka how are those values computed so i can find the nth term?

[CP5670]

There are both recursive and explicit evaluations of the coefficients, but they are extremely complicated and themselves involve several gamma functions. However, there is a fairly simple series for the logarithm of the gamma function, of which taking the exponential yields direct gamma function values. (this can be derived directly from Binet's second formula) I also added in the above series with a few more terms on it:

[Cannikin]

Bleh, forget these stupid expressions. Here's a simple challenge: write out googolplex (10^(10^100))

[Petrarch of the VBB]

You fool, Graham's number is much higher than a googolplex!

[Falcon X]

triple integrals anyone?

[CP5670]

just use the dirichlet multiple integral reduction on those; it works well for most of the common regions and integrands.

[Cannikin]

Originally posted by Petrarch of th VBB
You fool, Graham's number is much higher than a googolplex!

I never said it was higher. I said it was a "simple challenge". And if it's so much smaller, write it out

[CP5670]

still big enough to eat up all of gamespy's webspace though...

[Joey_21]

< nevermind > this is too risky for my processor

[Goober5000]

That would be 100*100*100 = 1,000,000.  A lot less than a googolplex.

A googol is
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

And a googolplex is 10 raised to that.

EDIT: ARGH!  Why is the number all chopped up?