Linked Questions

13
votes
11answers
5k views

A problem of an infinite number of balls and an urn

I think that the following problem originated in a probability textbook : You have a countably infinite supply of numbered balls at your disposal. They are all labeled with the natural numbers {1,2,3,...
14
votes
2answers
2k views

Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?

Consider the operator $\frac D{e^D-1}$ which we will call "shadow": $$\frac {D}{e^D-1}f(x)=\frac1{2 \pi }\int_{-\infty }^{+\infty } e^{-iwx}\frac{-iw}{e^{-i w}-1}\int_{-\infty }^{+\infty } e^...
38
votes
2answers
2k views

On the difference between two concepts of even cardinalities: Is there a model of ZF set theory in which every infinite set can be split into pairs, but not every infinite set can be cut in half?

An interesting question has arisen over at this math.stackexchange question about two concepts of even in the context of infinite cardinalities, which are equivalent under the axiom of choice, but ...
8
votes
3answers
1k views

Uses of Divergent Series and their summation-values in mathematics?

This question was posed originally on MSE, I put it here because I didn't receive the answer(s) I wished to see. Dear MO-Community, When I was trying to find closed-form representations for odd zeta-...
12
votes
3answers
1k views

"Values" of divergent integrals

Are there existing theories of integration in which $I_0 = \int_0^{\infty} dx$ and $I_1 = \int_0^{\infty} x \ dx$ are well-defined infinite elements in a non-archimedean extension of the reals? I can ...
-1
votes
2answers
611 views

Were there attempts to express derivatives of Delta function as polynomials of Delta function?

Is seems to me that it makes sense to presume some relations between derivatives of Dirac delta functions and its powers. I wonder, whether someone proposed a similar theory? Particularly, it could ...
13
votes
1answer
925 views

Is there an algebra for divergent series summation operators?

Let $D$ denote a divergent series and let $C$ denote a convergent series. Furthermore, let $s : $ { Series } $\to$ $\mathbb{C}$ be a regular, linear divergent series operator, which is either one ...
4
votes
2answers
700 views

Is there a sensible way to regularize $\int_0^\infty \tan x\, dx$?

I wonder if there is any sensible generalization of regularization which would be able to ascribe finite values to $\int_0^\infty \tan x \,dx$ and $\int_{-\infty}^0 \psi(x)dx$? Perticularly, since $\...
14
votes
2answers
816 views

Formula for volume of $n$-ball for negative $n$

Does the expression $$\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}R^n,$$ which gives the volume of an $n$-dimensional ball of radius $R$ when $n$ is a nonnegative integer, have any known ...
4
votes
3answers
364 views

Meaning of divergent integrals

In quantum field theory, one usually encounters divergent integral when one calculates loop functions, such as the integral $\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-m^{2})^{2}}$ which is divergent. ...
1
vote
1answer
271 views

Are there algebras over reals besides complex numbers, where identities, analoguous to $(-1)^i=e^{-\pi}$ and $i^i=e^{-\pi/2}$ hold?

Are there algebras over real numbers (with exponentiation), such that there is such $z$ that does not include components in $\mathbb{C}$ (or in a subset isomorphic to $\mathbb{C}$), for which $(-1)^z\...
1
vote
1answer
121 views

What are the properties of this set of infinite matrices and operations on them?

Consider infinite matrices of the form $$\left( \begin{array}{ccccc} a_0 & a_1 & a_2 & a_3 & . \\ 0 & a_0 & a_1 & a_2 & . \\ 0 & 0 & a_0 & a_1 & . \\ ...
1
vote
1answer
409 views

What are the consequences if we could express tangent via logarithm in an algebraic system? [closed]

Working on an algebra of divergent integrals I came to the following relation: If $\tau=\int_0^\infty dx$ then $$\ln (\tau+a)=\int_{0}^\infty \psi'(x+1/2+a)dx$$ and this directly gives the following ...
3
votes
0answers
379 views

What intuitive meaning "determinant" of a divergency (divergent integral or series) can have? [closed]

I am working on the algebra of "divergencies", that is, infinite integrals, series and germs. So, I decided to construct something similar to determinant of a matrix of these entities. $$\...
2
votes
1answer
302 views

A set of divergent integrals that I think, equal to $-\gamma$

So, we take $\frac{\text{sgn}(x-1)}{x}$ and apply $\mathcal{L}_t[t f(t)](x)$ four times. The transform is known to keep area under the curve. These integrals, I think, are equal to minus Euler-...

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