# Linked Questions

21 questions linked to/from An algebra of "integrals"

**13**

votes

**11**answers

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

**2**answers

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

**2**answers

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

**3**answers

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

**3**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**2**answers

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

**3**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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-...