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Questions tagged [divergent-integrals]

Divergent integral if the associated limit either doesn’t exist or is (plus or minus) infinity.

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Ratio of two diverging integrals

Consider the ratio: $$ r = \frac{\displaystyle\int_{-\infty}^{\infty}dx\, e^{-x^2 / 2a} x^2}{\displaystyle\int_{-\infty}^{\infty}dx\, e^{-x^2 / 2a}} $$ For $a > 0$ we have $r = a$ because after ...
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DiracDelta/x = -DiracDelta'? - Use and Correctness of Statement [duplicate]

One property of the Dirac Delta Distribution is $x \delta'(x) = -\delta(x)$ because of $\int x \delta'(x) f(x) dx = -\int \delta(x) (xf(x))' dx = -\int \delta(x) (f(x)+xf'(x)) dx = -\int \delta(x) f(x)...
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Which functions diverge in the following integral?

We have an integral of a real valued $C^2$ function on real numbers. The function is also monotonically increasing, and $f(0)=0$. Let $0<a<1.$ The integral: $$ I= \lim_{s \to 0} \int^{a}_s \left(...
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How to go from integral with $\theta((k^+)^2 - \vec{l}_\epsilon^2 - \vec{k}_\perp^2)$ to $\theta(k^+ - |\vec{k}_\perp|)$

I am trying to reproduce the calculation of the so called collinear-soft function, defined in arxiv 1410.6483. More concretely I would like to know the in between steps of the following equation since ...
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Tricky "Divergent" Integral: Correction to Groundstate

I am trying to rederive the results presented in the paper, in particular equation (30). That is, I am trying to compute the correction to the ground-state energy of a dipolar condensate due to beyond-...
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Two hard problems of improper integrals

I have 2 problems that I have been stuck on: Check the absolute convergence and convergence of these improper integrals: a) $\int\limits_0^\infty x^p\sin(x^q)dx$ $(q\neq 0)$. b) $\int\limits_0^\infty\...
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Improper integral converges or not [duplicate]

Find all the values $\alpha \in(0,\infty)$ such that the improper integral $$\int\limits_0^\infty \frac{\Bbb dx}{1+x^{\alpha}\sin^2x}$$ is convergent. My attempt is to analyze the cases (i) $\alpha =1$...
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Convergent integral

Find all the value of $\alpha >0$ such that $\int\limits_0^\infty \dfrac{\sin x}{x^\alpha +\sin x}dx$ converges. My attempt is to check the convergence of $I_1= \int\limits_0^1 \dfrac{\sin x}{x^\...
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Introducing a new constant $\alpha$ this way, what will be its properties? [closed]

Introduction First of all, let us introduce a concept of numerocity of a set, starting from the subsets of integers. $N(S)=\sum_{k=-\infty}^\infty p_s(k)=p_s(0)+\sum_{k=1}^\infty p_s(k)+\sum_{k=1}^\...
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With what classes of functions the equality $\int_0^\infty f(x)\,dx=\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx$ leads to paradoxes?

The following operators keep the area under the convergent integrals unchanged: $$\int_0^\infty f(x)\,dx=\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx=\int_0^\infty\frac1x\mathcal{L}^{-1}_t[ f(t)](x)\,...
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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 ...
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In umbral calculus, what is the established value of $\operatorname{eval}\ln (B+1)$?

In umbral calculus, what is the established value of evaluation (index-lowering operator) of the logarithm of $B+1$ where $B$ is Bernoulli umbra? In this preprint the author argues it to be $-\gamma$, ...
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If a double integral is bounded below by a $+\infty$ double integral, is it also $+\infty?$

If we have two functions $f,g$ defined on a region $R$ such that $f(x,y) \leq g(x,y),$ for all $(x,y) \in R,$ we can guarantee that $$ \iint_R f(x,y) dA \leq \iint_R g(x,y)dA. $$ This is a well-known ...
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General question regarding convergence of double integrals.

While studying double integrals, a student is certainly presented with the following property: Property. Given two functions $f$ and $g$ that are integrable over the rectangular region $R$ (region ...
xyz's user avatar
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what is the one-dimensional counterpart to the green-gauss theorem?

Are my answers to a and b correct? a) In a three-dimensional situation, the spatial variation of a scalar field is given by the gradient. What is the one-dimensional counterpart? Answer:The derivative ...
Mansoor Rahami's user avatar
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2 answers
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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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Showing that $\int_{0}^{\pi/2}\frac{\tan x}{1+e^x}\,dx$ diverges.

I want to show that the improper integral $\displaystyle\int_{0}^{\pi/2}\frac{\tan x}{1+e^x}\,dx$ diverges. I know that from $0$ to $\pi/2$ the following inequalities hold: \begin{align*} \frac{\tan ...
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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} \...
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Defining higher order (hypersingular) integrals in terms of derivatives of the Cauchy principal value?

In A Generalization of the Cauchy Principal Value, the author presents a way to assign values for hypersingular integrals of the form $$ I=\int_a^b\frac{f(x)\,\mathrm dx}{(x-u)^n},\quad u\in(a,b) $$ ...
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Why does the integral $\int_{1}^{\infty} x^p \,dx$ diverge for values of $p ≤ 1$?

$$ \int_{1}^{\infty} x^p \,dx $$ For the above integral, it was defined in class that this diverges for $$p \leq 1$$ but if $p$ is $0$, then wouldn't that mean $1/1$ which is $1$ and thus a number, ...
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inequivalence of Fourier transform?

I am working with the following function \begin{equation}\tag{1} {\cal F}\left(\textbf{k},\omega\right)=\frac{1}{i\omega+{\cal D}\left|\textbf{k}\right|^{2}} \end{equation} with $\textbf{k}=(k_x,k_y)$,...
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Divergent integral using contour

I have this integral where the integrand is divergent at $p=\{0, \pi i\}$, $$\lim_{r\rightarrow \infty} \int_{ib}^r \frac{1}{\cosh(p/2)\sinh(p/2)}\, dp$$ where $0 \leq b \leq \pi$ and $0 \leq r < \...
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Do these constants appear in other areas of mathematics?

I decided to consider divergent (to infinity) integrals as some new kind of number. Towards this end, I began by establishing certain rules defining equivalence of the integrals. It seems the usual ...
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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 ...
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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 $\omega\in\mathbb{C}$, $$ \lim_{\varepsilon\to0^+}\int_{C_\varepsilon}\frac{e^{i\alpha\omega}}{\omega^...
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5 votes
2 answers
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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's user avatar
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Can we assign meaningful values to $\int_{-a}^bf(x)/x^n\,\mathrm dx$ for $n=2,3,\dots$

The question title pretty much says it all. Let $f(x)\geq 0$ on $x\in[a,b]$ with $f(0)>0$ and $a,b>0$. In the case for $n=1$ we have the principal value intepretation which assigns a (usually ...
Aaron Hendrickson's user avatar
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1 answer
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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 ...
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Integral representation of $f(x)=0^x$

Recently I had an argument with Luboš Motl on Quora, where he had argued that $0^0$ should be left undefined in computer algebra systems, because $x^y$ has no limit at $(0,0)$ and $0^x=0$ at all $x>...
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Prove that Gamma function is divergent for $\Re(z)\leq 0$

How can I prove by definition that the Gamma function $$ \Gamma(z):=\int_0^{\infty}e^{-t}t^{z-1} \ dt $$ is divergent for $\Re(z)\leq 0$. I can prove it for real $z\leq 0$ only. Thanks!
jam's user avatar
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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). ...
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Can one represent divergent integrals or germs at infinity with surreal numbers?

I have been disliking the theory of surreal numbers for a while, but let's test it. So, we have a set of divergent improper integrals of continuous functions with the following ordering: $\int_0^\...
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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 ...
Anixx's user avatar
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1 answer
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How to determine that the improper integral $\int_{\pi}^{\infty}\frac{\sin^2(x)}{x}dx$ diverges? [closed]

How is the divergence of the improper integral $\int_{\pi}^{\infty}\frac{\sin^2(x)}{x}dx$ shown and determined with the improper integral comparison test or other methods?
Jack Herring's user avatar
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How to solve integrand that diverge?

In the following Eq. 1, the integral diverges. However, it is part of my equation for the Expectation of rate and it should not diverge. I will really appreciate any help as I am stuck on this for ...
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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? " ...
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Approximate the value of the intergral

Suppose that $\arctan(x)=\sum_{n=0}^{\infty} \frac{(-1)^nx^{2n+1}}{2n+1}$ for all x $\in$ $[-1,1]$. Use the least number of terms to approximate the value of the integral $$\int_0^{1/2} \frac{x-\...
Calamardo 's user avatar
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Why is it necessary to take$\int \frac{1}{x}~ dx$, and split it into two pieces, left and right side? Lets say its bounded by $2$ and $-2$?

The answer to this question is posted, but i don't understand what it means that as $b\to 0^+$ that this goes to infinity, and why is the integral split up in two?
AJT's user avatar
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1 vote
2 answers
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A divergent Definite Integral

I am trying the study a definite integral $$\int_{0}^{\pi}\frac{1}{(x-\frac{\pi}{2})^3+\cos{x}} dx$$ It is a divergent integral, but I am struggling to show that fact. Since the discontinuous point is ...
Jason Ng's user avatar
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-1 votes
1 answer
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Graphs of $e^{-y} = x$ and $y = \frac 1x$: Why do they seem so similar on $[0, 1]$, but one converges while the other diverges?

First things first, in the above graphs the one in blue is just a rotated version of ex from where the question did really form. I rotated it just in order to make comparisons easier. If we did the ...
GouravM's user avatar
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Is this integration by parts allowed?

I have a divergent integral of the following form: $$I = - \int_\mathbb{R} d\tau_3 \int_\mathbb{R} d\tau_4 \int_\mathbb{R} d\tau_5 \int_{\mathbb{R}^4} d^4 x_6\ \text{sgn} (\tau_{34})\ \text{sgn} (\...
Pxx's user avatar
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2 votes
1 answer
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Asymptotic estimation of a divergent integral

I am looking for an equivalent of the following integral : $$ \int_{0}^{A}{e^{t^\alpha}}dt $$ with $0<\alpha$ when $A$ tends to infinity. I call an equivalent a simpler function $f$ of $A$ such ...
Drill_zoo's user avatar
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1 answer
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Does this integral converge or diverge?

I have the following integral: $$I= \int_{-\infty}^\infty d\tau_3 \int_{-\infty}^\infty d\tau_4\ I_{13} Y_{134}, \tag{1}$$ with: $$I_{12} := \frac{1}{(2\pi)^2 x_{12}^2} \qquad \qquad Y_{134} := \...
Pxx's user avatar
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2 votes
1 answer
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How can I compute this integral involving $\Gamma$-functions?

I would like to find a closed form for the following integral: $$I=\int_0^1 d\alpha\ \alpha^{\omega-5/2} (1-\alpha)^{-1/2} \int_0^\alpha d\beta\ \beta^{2\omega-3} (1-\alpha-\beta)^{5/2-2\omega} (\...
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Show divergence of $\int_{1}^{\infty} \frac{x+\sqrt{x}+7}{x^2+2x+1} dx$

$$\int_{1}^{\infty} \frac{x+\sqrt{x}+7}{x^2+2x+1} dx$$ How can I show that the following integral diverges by using the direct comparison test? Thanks in advance!
Ludwig von Drake's user avatar
2 votes
2 answers
105 views

How can I prove that $\int_1^2 \frac{\ln(x-1)}{\ln(x^2-1)}~\text{d}x$ diverges?

I am looking at the following integral: $$\int_1^2 \frac{\ln(x-1)}{\ln(x^2-1)}~\text{d}x$$ Which, according to WolframAlpha, diverges. The integrand has a vertical asymptote at $x=\sqrt2$ over ...
Descartes Before the Horse's user avatar
3 votes
1 answer
401 views

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's user avatar
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1 answer
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Do Divergent Integrals have a unique regularisation?

I know that the same question for divergent sums is false, but cannot find much on divergent integrals. For example, consider the following divergent integral for positive reals $a,b$: $\...
Jack Tiger Lam's user avatar
1 vote
1 answer
131 views

Regularization of integrals of the type $\int^1_0dx\frac{\ln^n(1-x)}{1-x}$

In many books, articles and theses dealing with perturbative QCD it is claimed that integrals of the form $\int^1_0dx\frac{\ln^n(1-x)}{1-x}$ for $n\in\mathbb{N}$ become finite when multiplying the ...
Thomas Wening's user avatar
1 vote
0 answers
139 views

Integral involving a Gaussian and a fraction.

This question is a generalization of An identity involving the incomplete Beta function. . Let $x\ge 0$ and $\epsilon_\pm \in (1,\infty)$. We consider the following integral: \begin{equation} {\...
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