/sci/ - Science & Math

>>7100381
I wouldn’t agree about the naturally.
One important observation is that

<span class="math">f(x+h)=\sum_{k=0}^\infty\frac{1}{k!}f^{(k)}(x)·((x+h)-x)^k[/spoiler]

<span class="math">=\sum_{k=0}^\infty\frac{1}{k!}(h\frac{d}{dx})^k f(x)[/spoiler]

<span class="math">:=\exp(h\frac{d}{dx})\,f(x)[/spoiler]

and therefore

<span class="math">\frac{f(x+h)-f(x)}{h}=\frac{\exp(h\frac{d}{dx})-1}{h}f(x)[/spoiler]

<span class="math">=\frac{\exp(h\frac{d}{dx})-1}{h}\frac{d}{dx}\int^x f(y)dy[/spoiler]

If you set up a context where you can invert the operator, then the integral is determined by finite and infinitesimal differences, and finally

<span class="math">\frac{h\frac{d}{dx}}{\exp(h\frac{d}{dx})-1}=1-\frac{1}{2}(h\frac{d}{dx})+\frac{1}{12}(h\frac{d}{dx})^2+…[/spoiler]

That is to say these numbers really naturally appear in the finite differences business.

When Planck started quantum mechanics 120 years ago, he discretized the spectrum of the black body and derived his famous radiation law,

http://en.wikipedia.org/wiki/Planck%27s_law

the formula from the
>science. it works, bitches!
xkcd comic

<span class="math">B_\nu(\nu, T) = \frac{ 2 h \nu^3}{c^2} \frac{1}{e^\frac{h\nu}{kT} - 1}[/spoiler]
Is it an accident that Plancks radiation formula is the Mellin transform of the Riemann zeta function?

<span class="math">\zeta(s) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{x^{s-1}}{e^x - 1} dx[/spoiler]

Spoiler: No.
The number 12 is not as random as 17, say.
1+1+1+1+... and 1+1-1+1-1+... wanting to be one over +-2 is just it being the expression from the second most simple formula for coeffients, and 1+2+3+... wanting to be one over 2·3 (times 1/2!) is just it being the third most simple formula for coeffients.

>>7100370
I'm moving back to Vienna in summer and for now it seems that I might be working, for some time, at a firm that specializes in lube and friction. But I can always sell that as non-equilibrium statistical physics.