# Questions tagged [cauchy-principal-value]

Computation of Cauchy principal values of integrals. May be tied in with contour integration, but should be separate from definite-integrals.

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### Cauchy-Type integral

Greeting, I have an integral to solve, it is $$\int_{-\infty}^{+\infty} \frac{g(x)}{x-z}dx$$ with $g(x)$ a smooth, continuous, positive function, and $z=z_r + iz_i$ a complex number. I saw that this ...
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### Expressing the Principal Value of 1/x

I'm a graduate student and taking a course in analysis and came upon this definition for the cauchy principal value for 1/x, can someone kindly explain how this equality comes about and perhaps ...
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### Principal Value of 1/x Proof

How can we formally show that the Cauchy principal value of the function y(x)=1/x is a distribution, I understand that a distribution is a continuous linear functional on spaces of test functions, but ...
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### Principal value integral of $\frac{f(x)}{x}$ if f is continuous in [-1,1]

Suppose $f : [-1,1] \rightarrow \mathbb{R}$ is continuous, show p.v. $\int_{-1}^{1} \frac{f(x)}{x}dx$ exists. I know that since $f(x)$ is continuous on the closed bounded interval $[1,1]$, f is then ...
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### Cauchy Principal Value - nomenclature question [closed]

The CPV is usually defined as 'a way to assign values to otherwise undefined integrals.' Why is it never considered as an integral in its own right that generalizes and extends the Lebesgue integral,...
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### Distributional convergence of integrals

Hi :) I'm studying distributions and I would like to know if I'm doing this right. Let's take an ordinary function $$f(x)=\int_0 ^\infty dk\ \sin(kx)$$ and we want to study the functional on a test ...
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### Cauchy Principal Values, Cesaro and Abel summation versus Expectation

This might be a little bit of a soft question... I'm curious about Cauchy Principal Values, Cesaro Summation, and Abel summation, etc, especially how they relate to "regular" convergence (say, ...
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### Using the Cauchy Principal Value along a differentiable contour?

I am looking at the following problem from Marsden and Hoffman: Let $f ( z )$ be analytic inside and on a simple closed contour $\gamma$. For $z_0$ on $\gamma$, and $\gamma$ differentiable ...
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### Relationship between cauchy principal value and integrability of a singular point

Under what conditions does an integral have a cauchy principal value and how is it related to an integral having an integrable singularity? E.g $$p.v \int_{-\delta}^{\delta} \frac{dz}{z} = 0.$$ If I ...
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### Cauchy Principal Value and Residue Theorem: Apparent Contradiction

The following formula is well known and I already understood one of its proofs: $$\frac{1}{x\pm i\epsilon} = \mathrm{CH} \frac{1}{x} \mp i\pi\delta(x),$$ where the limit $\epsilon \to 0$ is implied ...
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### Meaning of the principal part for an analytic limit (as opposed to an integral)?

I am slightly confused as to the meaning of 'the principle part' of a limit, specifically in relation to the Kramers-Kronig relations as derived here on page 61. The source uses the 'Principal part' ...
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### Cauchy Principal Value calculation

I am self-studying the residue theorem and its applications and I tried solving a problem which involves finding the principal value for an improper integral but I am not sure if my approach/answer is ...
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### How to solve this Complex Integral using poles?

I want to find the green's function of a free particle, which depends of the integral: $$I = \frac{1}{4\pi ²ir} \int^{+\infty}_{-\infty} \frac{ke^{ikr}}{E-\frac{\hbar²k²}{2m}+i\eta} dk\,.$$ Then, ...
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### Derivative of ln|x| is the principal value of 1/x. Distribution Theory.

I have been looking at the proof for $\frac{d}{dx}\ln|x|=p.v.(\frac{1}{x})$ in the context of distributions and I am having trouble understanding why in the second term after integration by parts the ...
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### A question about integration of a orthogonal function

I have trouble figure out this orthogonal integration: $$\Psi _{1}=sin(\frac{n\pi x}{a}),\Psi _{2}=cos(\frac{n\pi x}{a})$$ $$\int_{-\infty }^{+\infty}sin(\frac{n\pi x}{a})cos(\frac{n\pi x}{a})dx$$ ...
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### How to calculate the particle derivative of the principal value of this integral?

I'm reading the book Mathematical Physics, and here's a problem I couldn't figure out. Let $\varphi(x)$ be a test function, show that \begin{equation}\frac{d}{dt}\{P \int_{-\infty}^{\infty} \frac{\...
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### Principal value of $\int^\infty_{-\infty}\frac1{x^2}dx$: counter-intuitive?

In attempt to evaluate $$\text P\int^\infty_{-\infty}\frac1{x^2}dx$$ we consider $$\oint_C\frac{1}{z^2}dz$$ where $C$ is an infinitely large semicircle on the upper half plane centered at the origin, ...
### Evaluation of $\int_{0}^{\infty}\frac{\sin(bx)}{x^2}\mathrm{d}x$
I'm trying to evaluate the following integral: $\int_{0}^{\infty}r^2\cdot\sin\big(\frac{b}{r^3}\big)\mathrm{d}r$ I had to solve a similar integral with cosine rather than sine and it was helpful to ...