# 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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### 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 $$\tag{1} {\cal F}\left(\textbf{k},\omega\right)=\frac{1}{i\omega+{\cal D}\left|\textbf{k}\right|^{2}}$$ with $\textbf{k}=(k_x,k_y)$,...
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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!
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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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### 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 ...
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### 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: {\...
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