Linked Questions
24 questions linked to/from An algebra of "integrals"
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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:
...
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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 ...
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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 ...
3
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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 ...
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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 ...
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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.
...
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1
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What are the properties of this set of infinite matrices and operations on them?
Consider infinite matrices of the form
$$\left(
\begin{array}{ccccc}
a_0 & a_1 & a_2 & a_3 & . \\
0 & a_0 & a_1 & a_2 & . \\
0 & 0 & a_0 & a_1 & . \\
...
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Are there algebras over reals besides complex numbers, where identities, analoguous to $(-1)^i=e^{-\pi}$ and $i^i=e^{-\pi/2}$ hold?
Are there algebras over real numbers (with exponentiation), such that there is such $z$ that does not include components in $\mathbb{C}$ (or in a subset isomorphic to $\mathbb{C}$), for which $(-1)^z\...
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What are the consequences if we could express tangent via logarithm in an algebraic system? [closed]
Working on an algebra of divergent integrals I came to the following relation:
If $\tau=\int_0^\infty dx$ then
$$\ln (\tau+a)=\int_{0}^\infty \psi'(x+1/2+a)dx$$
and this directly gives the following ...
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A set of divergent integrals that I think, equal to $-\gamma$
So, we take $\frac{\text{sgn}(x-1)}{x}$ and apply $\mathcal{L}_t[t f(t)](x)$ four times. The transform is known to keep area under the curve. These integrals, I think, are equal to minus Euler-...
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Are the shapes of the $\mathbb{R}^2$ plane and a disk of infinite radius different? Or otherwise, why their areas differ by $\frac\pi{12}$? [closed]
The calculation of the area of the $\mathbb{R}^2$ plane depends on filtering used. I think, the most natural filtering is along the radius in polar coordinates:
$$S_{\mathbb{R}^2}=\int_0^\infty 2\pi ...
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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 $\{1,0,−1,0,1,\ldots\}$...
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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 ...
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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.
$$\...
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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 ...