# Questions tagged [divergent-integrals]

The divergent-integrals tag has no usage guidance.

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### Can anyone figure out the sign of the following definite integral? [duplicate]

$$
\int_0^1 B^{\frac{1}{1-\alpha + \alpha x}} x^{k - 1} \left(\frac{\alpha \log{\left(B \right)}}{(\alpha-k-1)^2} +\frac{1}{k} + \log{\left(x \right)} \right) dx,
$$
where $B\geq 2$, $\alpha \in (...

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0
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### Biot-Savart-like integral for a toroidal helix

The following problem originates from Physics, so I apologize if I will not use a rigorous mathematical jargon.
Let us consider a toroidal helix parametrized as follows:
$$
x=(R+r\cos(n\phi))\cos(\phi)...

6
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3
answers

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### How do I solve the following definite integral (preferably by an asymptotic method)?

$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$
Note: $\mu$ here is an extremely small constant.
I have tried:
Estimating the integral by ...

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0
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### Crazy conjecture about Bernoulli umbra and reference request

For years umbral calculus have fascinated me. Bernoulli numbers (which represent powers of Bernoulli umbra) are involved in many classic power series expansions.
Yet, it still remains mistery what ...

2
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2
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### Assigning values to divergent oscillating integrals

I have recently run into a number of divergent oscillating integrals in various contexts. Thus, I have been led to desire general methods for assigning values to divergent oscillating integrals. All ...

3
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0
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394
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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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votes

1
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171
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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 ...

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0
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### Intuitively, what makes Bernoulli umbra so similar to the zero divisors in split-complex numbers?

Notation. Here I will denote Bernoulli umbra (its moments are Bernoulli numbers $B_n$) as $B_-$, $B_-+1$ as $B_+$ (an umbra with moments being Bernoulli numbers except $B_1=1/2$).
I will denote the ...

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0
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### Do the equalities $\int_0^∞1dx·\int _0^∞1dx=2\int_0^∞xdx$ and $\int_0^∞e^xdx·\int_0^∞e^xdx=2\int_0^∞e^{2 x}dx-2\int_0^∞e^xdx$ make sense?

Previously I tried to define multiplication of divergent integrals, but my approach turned out to be umbral-like.
Now, I decided to define multiplication of divergent integrals in a Hardy fields-like ...

0
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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 ...

1
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0
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212
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### What's the regularized value of these divergent integrals: $\int_0^\infty \ln x \, dx$ and $\int_0^\infty \frac{\ln x}{x^2} \, dx$?

When playing with divergent integrals $\int_0^\infty f(x) \, dx$ and their transformations with operators $\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx$ and $\int_0^\infty\frac1x\mathcal{L}^{-1}_t[ f(t)...

1
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0
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65
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### There is a ring with multiplication. Can we find a formula for division based on formula for multiplication?

Studying divergent integrals, I found a good formula for their multiplication:
$\int_0^\infty f(x)dx\cdot\int_0^\infty g(x)dx=\int_0^\infty D^2 \Delta^{-1} \left(\Delta D^{-2}f(x)\cdot\Delta D^{-2}g(x)...

6
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1
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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 ...

2
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### Hypermodulus and what mathematical objects have it

When researching divergent integrals, I decided to introduce a concept of "modulus" or "determinant" of divergent integral (and series). Basically, it is the exponent of the real ...

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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 ...

2
votes

1
answer

446
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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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0
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104
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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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0
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230
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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 ...

2
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172
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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 ...

0
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1
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419
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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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65
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### 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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2
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284
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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,
$\int_0^\infty (x^k-(k+1)!x^{-(k+2)})dx=0,$
...

3
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0
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408
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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.
$$\...

2
votes

1
answer

282
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### 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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### 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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2
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### 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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0
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171
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### 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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1
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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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1
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342
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### 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 ...