Linked Questions

1 vote
2 answers
591 views

Generalized limits

Cross-posted from Math SE. The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. It asks the following question: ...
user76284's user avatar
  • 1,793
-1 votes
2 answers
790 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 ...
Anixx's user avatar
  • 9,306
13 votes
1 answer
1k 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 ...
Max Muller's user avatar
  • 4,485
3 votes
1 answer
179 views

A suggestion for a superlimit

I have a question about summation methods. A value is assigned to a divergent sum. All methods agree that $\texttt{super-}\sum_{k=1}^{\infty} k^p = -\frac{B_{p+1}^{+}}{p+1}$ where $B_{p}$ are ...
Menno van der Ploeg's user avatar
2 votes
1 answer
234 views

List of assigned values of divergent series

I'm hoping to find a list of divergent sums where the assigned value is generally accepted. For instance $\sum_{n=0}^\infty (-1)^n$ is generally accepted to be $\frac{1}{2}$. Moreover, its agreed upon ...
Caleb Briggs's user avatar
  • 1,662
2 votes
1 answer
180 views

Integral transformation, Laplace-like

Is the following integral transformation of $f$ known (for suitable $f$ and $s\in\mathbb{C}$)? $$ \int_1^\infty f(t) \frac{e^{-ts}}{1-e^{-ts}}dt $$ It resembles somewhat the Laplace transformation. ...
borntomath's user avatar
1 vote
1 answer
177 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 & . \\ ...
Anixx's user avatar
  • 9,306
1 vote
1 answer
378 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\...
Anixx's user avatar
  • 9,306
1 vote
1 answer
415 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 ...
Anixx's user avatar
  • 9,306
0 votes
1 answer
418 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-...
Anixx's user avatar
  • 9,306
-1 votes
1 answer
228 views

Are the shapes of the $\mathbb{R}^2$ plane and a disk of infinite radius different? Or otherwise, why their areas differ by $\frac\pi{12}$? [closed]

The calculation of the area of the $\mathbb{R}^2$ plane depends on filtering used. I think, the most natural filtering is along the radius in polar coordinates: $$S_{\mathbb{R}^2}=\int_0^\infty 2\pi ...
Anixx's user avatar
  • 9,306
-2 votes
1 answer
382 views

What is Bernoulli umbra philosophically?

Well, Bernoulli umbra is an umbra whose moments are the Bernoulli numbers. But what is it philosophically? For instance, we can consider imaginary unit $i$ an umbra with moments $\{1,0,−1,0,1,\ldots\}$...
Anixx's user avatar
  • 9,306
3 votes
0 answers
369 views

Extending reals with logarithm of zero: properties and reference request

If we take logarithmic function, we can see that its real part at zero approaches negative infinity with the same rate and sign from any direction on the complex plane, while the Cauchy main value of ...
Anixx's user avatar
  • 9,306
3 votes
0 answers
407 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. $$\...
Anixx's user avatar
  • 9,306
1 vote
0 answers
99 views

What intuitive meaning "determinant" of a divergency (divergent integral, series, germ, pole or a singularity) can have?

I am working on the algebra of "divergencies", that is, infinite integrals, series, and germs. So, I decided to construct something similar to the modulus or determinant of a matrix of these ...
Anixx's user avatar
  • 9,306

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