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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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Why is the Lebesgue integral of $\frac{x}{1 + x^2}$ undefined

Related to Cauchy-random variables why is the Lebesgue-integral of the above (expectation) of such r.v not defined? $$\int_{\mathbb{R}} \frac{x}{\pi(1 + x^2)} = ?$$ It seems to me that in the ...
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Principal value of an integral

Let's consider the following integral: $$I(y)=\text{PV} \int_0^\infty \frac{f(x) dx}{x^2-y^2}$$ Where PV is the Cauchy principal value, $y \in \mathbb{R}, \qquad y>0$. It can be shown by ...
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Double integral (Cauchy principal value integral)

I am currently reading a book of Hilbert transforms and I have found with the following equality: $$\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{isx}\int_{-\infty}^{\infty^*}\frac{\phi(y)}{x-y}\text{ d}y\...
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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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Stochastic approximation of a Cauchy principle value integral.

Suppose I have a random variable $X\sim f_X(x|\boldsymbol\theta)$ with a well-defined expected value. The usual integral for an analytic solution of this expected value is $$\operatorname EX=\int_{\...
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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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Weak law of large numbers for reciprocal of normal

In two different journal articles: The First Negative Moment of Skew-t and Generalized Student's t-Distributions in the Principal Value Sense and The first negative moment in the sense of the ...
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Principal value and moving poles in nested integrals

I'm trying to understand how the following integral works: \begin{equation} I=\lim_{\eta\rightarrow 0_+}\int_{-\Lambda}^{+\Lambda} d\epsilon \int_{-\infty}^{+\infty} dx \frac{1}{x-a-i\eta\epsilon} \...
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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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How can we calculate the imaginary part of a fraction that has a term i0+ in the denominator (Sokhotski–Plemelj theorem)?

I have recently started dealing with thermal field theory for fermions and I am faced with a paper that, at some point, tries to calculate the imaginary part of a fraction that looks like: $$\frac{1}{...
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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, ...
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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 ...
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Contour for Principal Value Integral

I am trying to evaluate this principal value integral analytically $$ \mathcal{P}\int_0^\infty \mathrm{d}{x} \frac{1}{e^{x}-1}\frac{x}{x^2+\Omega^2}\frac{1}{x-\gamma} \,,$$ where $\gamma, \Omega \in \...
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Convergence of Parte Finie of $x \mapsto \frac{1}{x^2}$

I am working through some distribution theory notes, and was specifically working on this example Derivative of principal value distribution $1/x^2$ is equal to finite part distribution $-1/x^2$?. ...
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Solution for the Laplace Transform $\mathscr{L}\left\{\frac{t^{\alpha-1}}{t-\mu}\right\}(\beta)$

I have been looking for an explicit solution to the following Laplace transform for $\alpha,\mu,\beta>0$ \begin{equation} \frac{\beta^\alpha}{\Gamma(\alpha)}\mathscr{L}\left\{\frac{t^{\alpha-1}}{t-\...
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Conditional convergence of $\Bbb E[X^{-1}]$ for $X\sim\mathcal{N}(\mu,\sigma^{2})$ as $\operatorname{pr}(X>0)\to 1$

This question references the post Reciprocal of a normal variable with non-zero mean and small variance In the question the OP simulated observations from an inverse normal distribution, i.e. ...
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Lebesgue dominated convergence with a principal value integral

I have a sequence of functions $f_n(x) \rightarrow f(x) = \frac{g(x)}{x}$ pointwise almost everywhere such that $|f_n(x)| \leq \left|2\frac{g(x)}{x}\right|$ and the integral $\int 2\frac{g(x)}{x} ~dx$ ...
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Showing that $\int_0^{\infty} \frac{\cos(x)-e^{-x}}{x} dx = 0 $

I have to show that $$\int_0^{\infty} \frac{\cos(x)-e^{-x}}{x} dx = 0 $$ using a quarter circle in the upper positive plane and the function $f(z) = {e^{iz}}/{z}$. I think I have a solution where I ...
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Find Cauchy principal values of $\int_{-\infty}^{\infty}\frac{x}{(x^2+4)(x^2-2x+5)}\,dx$

I have been asked to find the Cauchy principal vlaues of the following problem using residues: $\int_{-\infty}^{\infty}\frac{x}{(x^2+4)(x^2-2x+5)}\,dx$ So far I have taken $\oint_C\frac{z}{(z^2+4)(z^...
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Problem with Hilbert Transform of $\operatorname{sgn}(t)$

I haven't found a duplicate question, but apologies in advance if this is a dup. At the DSP SE we were asked about the Hilbert Transform of the unit step (you math guys call it the Heaviside step ...
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How to show that $\mathscr{P} \int_{-\infty}^{\infty} \frac{d\omega}{\sqrt{\omega^2}} e^{ - i \omega t }$ is equal to $- \log(t^2)$

Fix $t>0$ and consider the following Principal Value integral: $$ \mathscr{P}\int_{-\infty}^{\infty}d\omega \frac{e^{-i \omega t}}{\sqrt{\omega^2}} = - \log(t^2) $$ This is the function that ...
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Using Contour Integration to Prove $\lim_{\epsilon\to 0^{+}}\int_{-1}^{1}x^{\epsilon-1}\,\mathrm{d}x=-\pi\mathrm{i}$

Let $\epsilon>0$. Then \begin{equation} L=\lim_{\epsilon\to0}\int_{-1}^{1}x^{\epsilon-1}\,\mathrm{d}x=-\pi\mathrm{i}. \end{equation} This can be proved by direct integration \begin{equation} L=\...
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PV of $\int_{-\infty}^\infty \frac{x^n~ e^{-ax^2} dx}{(x-x_1)(x-x_2)},~ (n=0,1,2,3,…)$

$$\tag{1} I_n = PV \int_{-\infty}^\infty \frac{x^n ~e^{-ax^2} dx}{(x-x_1)(x-x_2)},~ (n=0,1,2,3,...)~\text{with}~a>0~\text{and}~ x_1~x_2 \in \mathbb{R},$$ Is the principal value of $I_1$ ...
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An approach to evaluating a Cauchy principal value that yields unexpected extra imaginary term

$\newcommand{\PV}{\operatorname{PV}}$ I have been working on evaluating the first negative moment of a random variable with a piecewise density function by means of the Cauchy principal value, i.e. \...
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Approximating $I= \int\limits_0^1 \frac{1}{R^{2}-x^{2}} \exp (-\frac{a}{\sqrt{1-x^{2}}})\ dx$

I am trying to approximate the integral of the form: $$I= \int\limits_0^1 \dfrac{1}{R^{2}-x^{2}} \exp \left(-\frac{a}{\sqrt{1-x^{2}}}\right)\ dx $$ in the limits, $$a \ll 1 \ \ ; \ \ a \gg 1 \ \ \...
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Discontinuity across the cut of a function with a $\frac{1}{\sqrt{x^2-a}}$ factor

I Got this function: $$F(x)=\frac{1}{\sqrt{x^2-a}}\int_{-1}^1d\omega \frac{\sqrt{\omega^2-a}}{(x-\omega)}\phi(\omega)$$ $\phi$ is a well-behaved function and $a$ is a positive real number. I want to ...
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Find the principal value of $\big[\frac{e}{2}(-1-\sqrt3i)\big]^{3\pi i}$

I have been asked to find the principal value of: $\big[\frac{e}{2}(-1-\sqrt3i)\big]^{3\pi i}$ The textbook Complex Variables with Applications by A. David Wunsch only provides answers to the odd ...
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Existence of the Cauchy PV in the definition of logarithmic integral

I have some trouble trying to prove that the Cauchy principal value of this important integral $$ \int_0^x \frac{dx}{\ln x} $$ exists if $ x > 1 $. I thought about the expansion $$ \frac{1}{\ln(1+...
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Principale value of complex integral

I'm trying to calculate an integral of the form: $$ \textrm{Int}=PP\int_{1}^{\infty} \dfrac{x ^2 a^2 + 2 a x+ 2}{\sqrt{x ^2-1} \left[\left(b^2-c^2\right)x ^2 -b^2\right]} e^{-ax}\, dx $$ where $a&...
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689 views

Principal value of 1/x

I'm trying to show that p.v.$\frac{1}{x}(\varphi):= \lim_{\epsilon \to 0^{+}}\int_{|x|> \epsilon} \frac{\varphi(x)}{x}dx, \varphi \in \mathcal{S}$ exists and defines a tempered distribution, where $...
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How to Calculate PV$\int_0^{\infty}\frac{\cos ax}{x^4-1}\:dx\: $for all a in R [closed]

Can anyone help calculate this integral? $$\text{PV}\int_0^{\infty}\frac{\cos ax}{x^4-1}\,dx$$ for all $a \in \mathbb{R}$. Thanks in advance!
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Evaluation of $\int_{-\infty}^{\infty} \frac{\sin(x)}{x^n} \,\mathrm{d}x $?

The cauchy principal value of $ \displaystyle\int_{-\infty}^{\infty}\dfrac{\sin\left(x\right)}{x^n}\,\mathrm{d}x $ is defined as : $$ \begin{cases} {\displaystyle A=0\,,\quad n\in \mathbb{N^*}} \\[...
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Help understanding the weak topology on the dual of the Schwartz space?

I was working on the following problem, Prove that $\lim_{\epsilon \downarrow 0} \frac{x - x_0}{(x - x_0)^2 + \epsilon^2} = \mathcal{P}(1/(x - x_0)$, in the weak topology on $\mathcal{S}'(\mathbf{...
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Asymptotics of an integral $\mathscr{P} \int_{0}^{\infty} \frac{d\omega}{e^{\omega} - 1} \frac{\omega}{\omega^{2} - x^{2}}$ involving principal value

Consider the following integral for $x > 0$: $$ F(x) \ = \ \mathscr{P} \int_{0}^{\infty} \frac{d\omega}{e^{\omega} - 1} \frac{\omega}{\omega^{2} - x^{2}} $$ The $\mathscr{P}$ is there because ...
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What does the principal value $\mathscr{P}$ mean exactly in this integral?

I've encountered the following integral while reading a paper: $$ F(u,v) = \int_0^\infty \frac{d\omega}{e^{\omega} - 1} \mathscr{P}\left\{ \ln\left( \frac{ \left( u - 2 \omega \right)^2 \left( v + 2 ...
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Cauchy Principal Value and Divergent Integrals

So, I've been wondering why PV integrals don't prove convergence of Integrals? For example, this integral $$ \int_{-1}^1 \frac{1}{x} \mathrm dx$$ converges to 0 using Cauchy Principle, and diverges ...
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Bose-Einstein function as real part of polylogarithm: $\overline{G}_{s}(x)= \Re \mathrm{Li}_{s+1}(e^x)$

For real $x<0$ the Bose-Einstein integral of order $s$ is given at https://dlmf.nist.gov/25.12.E15 as $$G_{s}(x)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{e^{t-x} -1}\mathrm{d}...
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Reducing this principal value integral to something I can evaluate numerically

I have the following principal value integral: $$\mathcal{P}\int_0^\infty\frac{x^4}{\left(1+\frac{x^2}{B^2}\right)^{4}\sqrt{C^2+x^2}\left[\sqrt{C^2+a^2}-\sqrt{C^2+x^2}\right]}dx$$ where $a,B,C\in\...
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rate of limits in Cauchy's principle value

For Cauchy's principal value, one defines the principle value as \begin{gather} PV\int_a^b f(x) dx = \lim_{\epsilon \to 0^+} \left[ \int_a^{c-\epsilon}f(x) dx + \int_{c+\epsilon}^b f(x) dx \right] \...
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Taking a limit on a one sided integral of a complex exponential

Some lecture notes of mine quote this apparently well known result: $I(\omega)=\lim_{\eta\rightarrow0^+}\int_0^\infty e^{(i\omega-\eta)s}\text{d}s=\pi\delta(\omega)+i\mathcal{P}(\frac{1}{\omega})$, ...
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Principal value integral $\frac{f(x)}{(x-y)^2}dx$ in $\mathbb{R}$

I wish to compute Cauchy principal value integral $$P\int_0^1\frac{f(x)}{(x-y)^2}dx$$ numerically, but using the PV prescription $$P\frac{1}{x^2}=\frac{1}{2}(\frac{1}{(x+i \epsilon)^2}+\frac{1}{(x-i\...
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190 views

how to deal with principal value integral qns that have three poles?

There is this question which ask me to calculate the principal integral $$ \mathcal{P} \int_{-\infty}^\infty \frac{e^{-ix}}{(x+1)(x^2+1)} dx \, . $$ I find out that the poles are $x=-1$, $x=-i$ and $...
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What makes the Cauchy principal value the “correct” value for a integral?

I haven't been able to find a good answer to this searching around online. There is a related old question here, but it never received much attention. Suppose I have some physical property that I ...
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193 views

Half-range Fourier transform of $e^{-ixt}$

I'm reading a textbook that has the following sentence: Making use of the formula $$\int_0^\infty dt e^{-i x t} = \pi \delta (x) - i \text{P}\frac{1}{x} , $$ where $\text{P}$ denotes the Cauchy ...