Questions tagged [divergent-integrals]
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An operation is defined on polynomials. How do I generalize it to other classes of functions?
I am currently researching divergent integrals.
Definition. An extended number is an expression of the form $\int_a^b f(x)\,dx$, where $a,b\in \overline{\mathbb{R}}$ and function $f(x)$ is defined ...
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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 ...
2
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1
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Where do these divergent integrals appear in mathematics and physics?
I have already asked a similar question, albeit far more extensive, but it was criticized and closed for being too extensive and promotional. So, here is a greatly truncated and focused version.
Since ...
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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 ...
2
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Regularization of the area under hyperbola
So, I am trying to find the regularized value of the divergent integral $I=\int_1^\infty \sqrt{x^2-1}dx$. Since the area of $\int_0^1 \sqrt{1-x^2}dx=\frac\pi4$, I wonder whether the area under ...
2
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Did anyone ever propose the distinction between "divergent to infinity" as opposed to "divergent but with finite average"?
There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences.
Still, in my view there is fundamental difference between divergent ...
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Can we meaningfully ascribe values to these divergent integrals?
My gut feeling is that
$\int_0^\infty (1-\frac1{x^2})dx=0$
$\int_0^\infty (x-\frac2{x^3})dx=0$
$\int_0^\infty (x^2-\frac6{x^4})dx=0,$
etc, and in general,
$\int_0^\infty (x^k-(k+1)!x^{-(k+2)})dx=0,$
...
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How is this expression for the regularization of integrals of monomials, given in a paper, justified? How strong is argument in favor?
In this answer by Carlo Beenakker he cites the following regularization formula:
$$\int_0^\infty x^p\,dx\mathrel{"="}\frac{(-1)^{p+1}}{(p+1)(p+2)},\;\;p=0,1,2,\dotsc,$$
citing Tafazoli - Calculation ...
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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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Generalization of Levi-Civita type construction towards divergent integrals and corresponding questions
A known generalization of Levi-Civita field is a field of Hahn power series of $\varepsilon$ of the form
$\mathbb{R}[[\varepsilon^{\mathbb{Q}}]]$. Assuming $\varepsilon=1/\omega$, we can naturally ...
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Is there a formula or algorithm to remove infinitesimal and oscillating parts from an expression while keeping finite and infinite ones? [closed]
Below, we interpret divergent integrals as germs of partial integrals at infinity:
$$\int_0^\infty f(x) dx=\operatorname{bigpart} \int_0^\omega f(x) dx$$
where $\operatorname{bigpart}$ means taking ...