Divergent integral if the associated limit either doesn’t exist or is (plus or minus) infinity.
Questions tagged [divergent-integrals]
64 questions
9
votes
2 answers
Small parameter expansion of the definite integral, divergence of coefficients
I’m considering the following integral with one parameter $\omega$
$$I(\omega):=\int_0^{\infty}(1+x)^2\Bigg(\sqrt{x^2+\frac{\omega^2}{(1+x)^\delta}}-x\Bigg)dx,$$
where $\delta>2$ and $\omega $ is small parameter.
I want to know the expansion of…
Siam
- 233
8
votes
1 answer
regularization of a divergent integral
is there any way to regularize the following divergent integral :
$$\int_{0}^{\infty}\frac{dx}{xe^{x}(e^{x}-1)}$$
the integral comes from trying to find an analytic continuation of
$$I(s)=s\int_{0}^{\infty} \frac{dx}{2x}\left(E_{s/2}((\pi…
Mohammad Al Jamal
- 575
- 1
- 16
- 28
6
votes
2 answers
Are there any good examples from other math fields or intuition supporting $\int_0^1\frac1xdx=\int_1^\infty\frac1xdx$?
This question is related to the potential possibilities of classification of divergent integrals more precisely than just "divergent to infinity" and the like. Improper divergent integrals can have many distinguishing properties, like germ at…
Anixx
- 10,161
5
votes
2 answers
Fourier transforms: divergent integrals?
I am studying different integral transform methods, and I am confused on why saying things such as
$$
\mathcal{F}^{-1}[1] = \delta(x)
$$
is valid? If you actually plug this in,
$$
\mathcal{F}^{-1}[1] =…
Riemann'sPointyNose
- 6,753
4
votes
1 answer
What is the Fourier transform of $\text{csch}^2(t)$?
Mathematica indicates the Fourier transform of $\text{csch}^2(t)$ is
$$\mathcal{F}_t\left[\text{csch}^2(t)\right](\omega)=-\frac{\pi \omega \coth\left(\frac{\pi \omega}{2}\right)+2}{\sqrt{2 \pi}}\tag{1}$$
but also indicates
$$\mathcal{F}_{\omega…
Steven Clark
- 8,744
4
votes
1 answer
Why is $\ln 0\ne-\ln \infty$?
The title of this post is intentionally sensational, but what I am really going to do is to compare the divergent integrals $\int_0^1\frac1xdx$ and $\int_1^\infty\frac1xdx$.
Let's consider the transform $\mathcal{L}_t[t f(t)](x)$. It is notable by…
Anixx
- 10,161
4
votes
3 answers
How do you "regularize" infinite integrals?
This question was inspired by the post:
" Is there a solid reason why some people assume the fundamental theorem of calculus should still hold for divergent integrals with improper bounds? " (and the follow-up discussion). Long story short, the OP…
user700480
3
votes
2 answers
Confusion regarding a second order pole on the contour
Let us define a function
$$f(x) = \frac{1}{(x^2 + A^2) (x^2 - B^2)^2}$$
where, $A,B > 0$. We need to calculate
$$G = \int_{-\infty}^{\infty} f(x) dx $$
I am confused regarding whether $G$ diverges or not.
Argument 1: $G$ does not diverge. By complex…
Prem
- 187
3
votes
2 answers
Extending reals with logarithm of zero: properties and reference request
If we take logarithmic function, we can see that its real part at zero approaches negative infinity with the same rate and sign from any direction on the complex plane, while the Cauchy main value of the imaginary part averaged over any circle…
Anixx
- 10,161
3
votes
0 answers
What are the properties of this new characteristic of mathematical objects?
I will call it "hypermodulus". In simple words, hypermodulus is the exponent of the scalar part of the finite part of the logarithm of the object: $H(A)=\exp (\operatorname{scal} \operatorname{f.p.} \ln A)$.
The "finite part" is meant to mean…
Anixx
- 10,161
3
votes
0 answers
Does the Euler-Mascheroni constant $\gamma$ correspond to infinite hyperbolic angle?
So, I am trying to find the regularized value of the divergent integral $I=\int_1^\infty \sqrt{x^2-1}dx$. Since the area of sector of a circle $\int_0^1 \sqrt{1-x^2}dx=\frac\pi4$, I wonder whether the area under hyperbola would be interesting as…
Anixx
- 10,161
3
votes
1 answer
Contour integral $\int_{C_\varepsilon}\frac{e^{i\alpha\omega}}{\omega^2}\mathrm{d}\omega$
One intermediate step of an exercise requires to evaluate the the following integral with variable…
zytsang
- 1,553
- 1
- 16
- 16
3
votes
3 answers
Interpreting the logarithm as a sum of simple poles along the negative real axis
I've heard it remarked that you can basically consider $\log z$ to be a function which has simple poles everywhere on the negative real axis (with a constant "residue density" at each pole). This would be something like
$$ \log z = \int_0^\infty…
sasquires
- 1,685
- 12
- 17
3
votes
1 answer
Divergent Integral?
Why is $$\int_{-\infty}^{\infty} \frac{2x}{1+x^2}dx$$
divergent, when the function being described is clearly an odd function and $$\int_{-a}^{a} \frac{2x}{1+x^2}dx = 0$$
for any finite a?
wheelix
- 307
2
votes
0 answers
Asymptotic expansion double integral
this question is for experts in asymptotic analysis, in particular expansions of integrals.
Consider the following type of integral
$$
I(\ell) = \ell^2\int_{[0,2\pi]^2}\frac{dk dq}{(2\pi)^2}…
gdvdv
- 106