Recent content by jostpuur

I realised soon after posting the question that I had made a mistake with the sets ]a,b[ and [a,b], but I thought it wouldn't matter much, and didn't start quickly editing it. Now it looks like that these details do matter after all. I got a feeling that the question could probably be made...

Suppose f:]a,b[\to\mathbb{R} is some differentiable function. Then it is possible to define a new function ]a,b[\to [a,b],\quad x\mapsto \xi_x in such way that f(x) - f(a) = (x - a)f'(\xi_x) for all x\in ]a,b[. Mean Value Theorem says that these \xi_x exist. One question that sometimes...

The first look at a scattering process is something like this: We define an initial state |\textrm{in}\rangle = \int dp_1dp_2 f_{\textrm{in,1}}(p_1) f_{\textrm{in,2}}(p_2) a_{p_1}^{\dagger} a_{p_2}^{\dagger} |0\rangle Here f_{\textrm{in,1}} and f_{\textrm{in,2}} are wavefunctions that define...

Yes, this is the same series whose first terms I wrote down with their more explicit values. Very nice, thanks.

By using the formulas \sin(\alpha)\sin(\beta) = \frac{1}{2}\big(\cos(\alpha - \beta) - \cos(\alpha + \beta)\big) \sin(\alpha)\cos(\beta) = \frac{1}{2}\big(\sin(\alpha - \beta) + \sin(\alpha + \beta)\big) it is possible to write the powers (\sin(x))^n in a form where non-trivial powers do...

Is there a reason for why you don't deal with the original integral via calculus of residues?

The input x-1 is an unnecessary complication for the actual result, and we could also state that the formula D_x \arctan\big(x + \sqrt{1 + x^2}\big) = \frac{1}{2}\frac{1}{1 + x^2} is true.

I see, thanks. If f reached negative values, then f' should have a third zero somewhere. You made a typo with the quantity \big(\frac{\pi}{2}x\big)^2 in an intermediate step.

What do you think about the claim that \frac{x}{\frac{1}{a} + \frac{x}{b}} \;<\; \frac{2b}{\pi}\arctan\Big(\frac{\pi a}{2b}x\Big),\quad\quad\forall\; a,b,x>0 First I thought that if this is incorrect, then it would be a simple thing to find a numerical point that proves it, and also that if...

I know a proof for a theorem that states that it is not possible to write the plane as a union of closed disks in such way that the interiors of the disks would be disjoint. In other words \mathbb{R}^2 = \bigcup_{i\in\mathcal{I}} \bar{B}(x_i,r_i) and i,i'\in\mathcal{I},\; i\neq...

The proof on math stack exchange contains a little mistake, but I got a feeling that the proof works anyway, and the little mistake can be fixed. The mistake is that the guy forgets that the sets G_k depend on epsilon, so they are like sets G_k(\epsilon). He first proves that...

Is it possible to use Axiom of Choice to prove that there would exist a sequence (A_n)_{n=1,2,3,\ldots} with the properties: A_n\subset\mathbb{R} for all n=1,2,3,\ldots, A_1\subset A_2\subset A_3\subset\cdots and \lim_{k\to\infty} \lambda^*(A_k) < \lambda^*\Big(\bigcup_{k=1}^{\infty}...

You can obtain some results concerning that question by examining the Fourier transforms. This approach suffers from the obvious shortcoming that not all functions have Fourier transforms, but anyway, it could be that Fourier transforms still give something.

For a moment I tried to get a rigorous lower bound for the two dimensional sum by starting from \frac{1}{x^2 + y^2} \leq \frac{1}{\lfloor x\rfloor^2 + \lfloor y\rfloor^2} but unfortunately a bunch of technical difficulties arise from the polar coordinates working nicely around the origin...

I guess the original question has already been answered, but anyway, I could not help taking a look at the function you just defined, and I guess of course you have to keep doing some exercises to maintain your math skills :-p The formula x\mapsto \frac{x^2 - \lfloor x^2\rfloor}{x} defines...