Lamé curve

[jheriko]

Originally posted by Atero
I said n/0 equals something.

i.e. n/0 is defined, therefore 1/x is continuous.

Also, +infinity and -infinity are real numbers (Sup(R) and Inf(R)). I'm not sure about 'natural infinity' since it seems not to fit in anywhere, or even to make sense in terms of real numbers.

Originally posted by Atero
General mathematical question: What is the (psuedo-)exact definition of a factorial? Cus I thought it was just n*(n-1)*(n-2)...*1=n!, which means that if n=0, it's 0*1=n!, right?

If that definition went down to zero then every factorial would be zero right? Anyway, what you are looking for is the gamma function. It is defined to be the extension of the factorial function into the field of complex numbers (and therefore reals, rationals and (the ring of) integers too). There are several ways to define and evaluate the function, for real numbers there are some fairly straightforward methods (especially for x>0) mostly integrals and infinite products if I can remember correctly. Here are a couple that I can remember:

I even bothered to make it look correct for you (shame that there is no subscript and superscript).

code:




for x>0.

        1

G(x) = ò (-log t)^(x-1) dt

        0



        ¥

G(x) = ò t^(x-1) e^-t dt

        0

Those two are actually the same thing just in a different form, the bottom one can often be easier to evaluate even though it involves infinity. (i'll explain integrals if you want but preferably not on the forums, it would require a colossal post)

Originally posted by Atero
Another general mathematical question: Is there a real mathematical definition of the absolute value function, or is it just |x| = +x?

z=a+ib
|z|=sqrt(a*a+b*b)

The modulus function is mainly for complex numbers. If b=0 then z is real, so |z| is real.

Originally posted by Atero
And if I'm not mistaken it does have an inverse funciton: f(x)=|x|; f^-1(x)=+-x

No single valued inverse, that isn't really what I would call an inverse function, multi-valued functions like that are useless since you will never know which value you need without already knowing it.

Originally posted by Atero
Which is why conics and Lame' curves are usually written with absolute value signs - when inverted, you use the positive-or-negative square root, instead of just the square root (which is always negative when written by itself).

I'd never seen a conic or any curve using |x| until we had this discussion. Its probably just convention, but it would definately cause problems when dealing with a general conic, for instance, where the curve is off centre and rotated, which is probably why I've never seen it since I was introduced to conics in the general form and then shown the simpler (specific) equations as a consequence of the general form.

Originally posted by Atero
What do you mean by 'the real thing?' If you're talking about the real formula, that IS the real formula, just re-arranged to fit scripting

I'm just assuming (like you are) that I am correct.