6 votes

Analytic continuation of convergent integral

Technically, your integral is not well-defined because the path goes through $z=1$; the remedy I see presently (unless you have a definition for the contour going through $z=1$), is to move the ...
Jack L.'s user avatar
  • 1,423
6 votes

Is there a sensible way to regularize $\int_0^\infty \tan x\, dx$?

Two pieces of good news. The formulae $\int_{0}^\infty \sin(x)\,dx=1$ and $\int_0^\infty \tan x\,dx=\ln 2$ hold. (Can‘t say anything about $\psi$ since I don‘t know what it means). These are not ...
user131781's user avatar
  • 2,442
6 votes

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

Here is a log-log plot of $$\delta I=I_{\text{appr}}-\int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx,$$ as a function of $\mu$, with $$I_{\text{appr}}=...
Carlo Beenakker's user avatar
5 votes

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

It is more or less straightforward to write down an asymptotic expansion around $\mu\to0$, $$ I(\mu)\sim \frac{2 \mu ^2}{15}-\frac{\mu ^4}{9}+\frac{\pi \mu ^5}{15}+\frac{1}{450} \mu ^6 (60 \log (\mu )...
AccidentalFourierTransform's user avatar
3 votes

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

I really like the double-series approach by @AccidentalFourierTransform, however I got remaining $\Lambda$s in the terms of order $O(\mu^8)$ onwards. Thinking about this approach, I located the ...
Fred Hucht's user avatar
  • 2,705
3 votes

Is there a sensible way to regularize $\int_0^\infty \tan x\, dx$?

Well, after some thinking, I came to the following method. To find $\int_0^\infty \tan x\, dx$ we have to subtract from the mean value of the antiderivative at infinity its value at $0$. This follows ...
Anixx's user avatar
  • 9,316
2 votes

Where do these divergent integrals appear in mathematics and physics?

Perhaps the most "famous" divergent integral in physics is the Casimir vacuum energy of the electromagnetic field, $$\int_0^\infty x^2\sqrt{1+x^2}\,dx"="-\frac{1}{10},$$ and the related ...
Carlo Beenakker's user avatar
2 votes

Is there a formula or algorithm to remove infinitesimal and oscillating parts from an expression while keeping finite and infinite ones?

This isn't an answer, just a long comment. If this is from an established field, and I'd guess it is, that needs to be part of the question. Not knowing one, I will blindly sally forth because I am ...
Aaron Meyerowitz's user avatar
2 votes

Can we meaningfully ascribe values to these divergent integrals?

Depends on how you choose to define it :) A natural way is to define $\int_0^{\infty} = \lim_{a\rightarrow0} \int_a^{1/a}$, in which case: \begin{eqnarray*} \int_0^{\infty}\left(1-\frac1{x^2}\right)dx ...
Ralph Furman's user avatar
  • 1,243
1 vote

Assigning values to divergent oscillating integrals

If you do not want to work within the framework of distribution theory, an alternative reading might be "Oscillatory Integrals" by Ioannis Parissis.
Alex M.'s user avatar
  • 5,207
1 vote
Accepted

Assigning values to divergent oscillating integrals

There is indeed such a method which is elementary and allows one to rigorously give a numerical value to such integrals, in particular, the value $1$ to $\int_0^\infty \cos x\, dx$. This uses two ...
order's user avatar
  • 26
1 vote

An operation is defined on polynomials. How do I generalize it to other classes of functions?

Answering my own question. SEE THE UPDATE BELOW The operator looks like this: $\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)\...
Anixx's user avatar
  • 9,316
1 vote

A set of divergent integrals that I think, equal to $-\gamma$

There might be a mistake here, but formally I'm getting \begin{align} \int_{x=0}^{x=\infty} \frac{\mathrm{sgn}(x-1)}{x} \mathrm{d}x &= \int_{x=1}^{x=\infty} \frac{1}{x} \mathrm{d}{x} - \int_{x=0}^{...
user76284's user avatar
  • 1,793
1 vote

Can we meaningfully ascribe values to these divergent integrals?

Warning: The following contains formal manipulations that ignore convergence. Proposition: \begin{align} \mathrm{regularized} \int_0^\infty \mathrm{d}x^s = 0 \end{align} for all $s$ such that $\Re(s) \...
user76284's user avatar
  • 1,793
1 vote
Accepted

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

Apparently, the discrepancy comes from my use of non-natural definition of multiplication of divergent integrals. When using a more natural and intuitive Levi-Civita field kind of construction, the ...
Anixx's user avatar
  • 9,316
1 vote

Is this relation between divergent intergals justifiable?

I don't see where the $\frac{1}{(n+1)!}$ factor comes from. If you take $II = \int_{a}^{b}\frac{1}{t^{n+2}}dt$ and perform the change of variables $t \to 1/x$, you get $$-\int_{1/a}^{1/b}\left(\frac{...
Jacques Carette's user avatar

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