Here's your own place to post that jazz and non-jazz.
![Laughing :lol:](./images/smilies/icon_lol.gif)
Dude, complex numbers are everything! XD the properties of the gamma function arise from it being defined starting from a exponential, and that's why 2*pi appears a lot. The zeta function is almost useless if it isn't defined over the complex plane. Also in statistics, the gauss bell's normalised by dividing by the square root of two*pi. Why? Because the canonical function is e^(-x^2/2), which integral over all the reals gives sqrt(2*pi). The area of a circle is the integral of its boundary with respect to the radius, that's why the 2 in the formula disappears. In formulas for higher dimensional spheres it stays. In group theory there's also an exponential map (you can exponentiate matrices!). The logarithm of a complex function is multivalued, the value that appears is 2pi*i and on it depends the cauchy theorem and all the holomorphic constructions. The differential equations often need complex numbers to work, and 2*pi appears on them as a consequence. But exponentials and logarithms are sometimes related with that constant, because some of its real properties depend on complex numbers. So if you try to delete everything related to complex numbers from your discussion, you end up with nothing (no sines, no cosines, very limited exponentials, unexplainable convergence theorems and restrictions,... and that's just the beginning)Taalit wrote:Maybe *you* have only seen the math where it's mostly 2pi. What do you think of that, huh?OnyxIonVortex wrote:Then you haven't seen enough mathematics yetTaalit wrote:2pi doesn't appear more than just pi, in pure mathematics. That's kinda only a physics thing.
Exhibit A: my signature.it appears a lot in the context of complex numbers, exponentials, logarithms, trigonometry, hiperbolas, spheres, fourier transforms, distributions, hermitian operators, extensions of complex numbers... I'd guess it's something like 95% 2*pi vs. 5% another constant*pi
Also, pi in general doesn't occur very often in logarithms or exponentials at all unless you're applying them to complex numbers, so you're kind of just giving an overall topic and then stating all of the things it includes. Also: What do you mean 'extensions of complex numbers'?
No it isn't, for integer values above 1 it gives you the probability of x integers being coprime. That's a pretty good use!OnyxIonVortex wrote: The zeta function is almost useless if it isn't defined over the complex plane.
But isn't pi always connected to circles (or spheres or whatever) in some peculiar and contrived way? If another constant makes more sense with circles, why use the other one at all?Taalit wrote:Fine, that is a lot of things. But you could still easily find a ton of other things that don't involve 2pi. My whole point is not that 2pi doesn't happen *often*, it's that other multiples also occur often so there's no real point or use to changing it.
No it isn't, for integer values above 1 it gives you the probability of x integers being coprime. That's a pretty good use!OnyxIonVortex wrote: The zeta function is almost useless if it isn't defined over the complex plane.
In that I do agree.Taalit wrote:Fine, that is a lot of things. But you could still easily find a ton of other things that don't involve 2pi. My whole point is not that 2pi doesn't happen *often*, it's that other multiples also occur often so there's no real point or use to changing it.
No it isn't, for integer values above 1 it gives you the probability of x integers being coprime. That's a pretty good use!OnyxIonVortex wrote: The zeta function is almost useless if it isn't defined over the complex plane.
Well, you can say mathematicians like simplicity, but they also have to deal with tradition and with a centuries-old established notation, and hundreds of papers written in that notation. It's about balance between those things, like in many other consensus-like subjects.But isn't pi always connected to circles (or spheres or whatever) in some peculiar and contrived way? If another constant makes more sense with circles, why use the other one at all?
EDIT: Well, I suppose you don't really accept that, so my argument is silly.