Questions tagged [divergent-integrals]

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Compute $\int dq \ q^{n} j_{\ell} \left( q r \right) j_{\ell}\left( q R \right)$

I am interested in analytical formulas for integrals of products of spherical Bessel functions times a power: $$I_{n,\ell}(r,R) \equiv \int_0^{\infty} dq \ q^{n} j_{\ell} \left( q r \right) j_{\ell}...
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207 views

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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252 views

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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113 views

Would the generalized ring of periods make sense?

The Wikipedia's article on ring of periods reads "a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain". But what if we include here ...
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43 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 ...
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144 views

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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74 views

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 ...
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1answer
226 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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25 views

Fractional power of the operator $\mathcal{L}_t[t f(t)](x)$ and equivalence of divergent integrals

I wonder whether an expression for fractional power of operator $\mathcal{L}_t[t f(t)](x)$ that involves Laplace transform can be derived? I am asking this because this operator preserves the area ...
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2answers
192 views

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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371 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
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1answer
206 views

Analytic continuation of convergent integral

I was trying to solve the following integral: $$I = \oint _{|z|=1}\frac{dz}{2 \pi i z}\int_{0}^{\infty} dr \dfrac{e^{-\tfrac{r^2}{z^2}}r^{2n+1}}{z^2(z-1)} $$ The singular structure in the $z$ ...
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124 views

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

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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690 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 $\...
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151 views

Prove the following property about natural integral

Natural integral is the distinguished antiderivative of a function that can be understood as an analytic continuation of consecutive derivatives of a function towards $-1$th order. It is defined as $...
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406 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 ...
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303 views

Is this relation between divergent intergals justifiable?

Graf's book on hyperfunction theory says (page $36$) that $$\frac1{(x-i0)^n}=\frac{(-1)^{n-1}\pi i}{(n-1)!}\delta^{(n-1)}(x)+\operatorname{fp}\frac1{x^n},$$ while the table of Fourier transforms ...