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 ...
- 1,453
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 ...
- 2,472
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
$$I_{\text{appr}}=...
- 188k
5
votes
How do I solve the following definite integral (preferably by an asymptotic method)?
It is more or less straightforward to write down an asymptotic expansion around $\mu\to0$,
$$
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 )...
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 ...
- 3,671
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 ...
- 10.1k
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 ...
- 188k
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 ...
- 30.1k
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 ...
- 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.
- 5,407
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 ...
- 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)\...
- 10.1k
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}^{...
- 2,203
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) \...
- 2,203
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 ...
- 10.1k
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{...
- 11.8k
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