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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1answer
22 views

Show $\operatorname{p.\!v.}\left(\frac{1}{x}\right)\,$ is an odd distribution

Let $\varphi$ be a test function (belonging to the set of smooth functions with compact support) We define the distribution principal value of $1/x$: $$\left\langle\operatorname{p.\!v.}\left(\frac{1}{...
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Why does this improper integral converge

I can't understand a part of a proof. Here is the strange part isolated. Let $1<q<2$. Then $(\int_{|z-\tau|\leq R}\frac{d\tau}{|z-\tau|^q})^q = (\frac{2\pi R^{2-q}}{2-q})^q$ The integral is over ...
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171 views

Value of the contour integral around an interesting singularity of $1/(e^z-1) \cdot 1/(e^{1/z}-1)$.

Let $f$ a function $f:\mathbb{C}\to\mathbb{C},$ $$f(z)=\frac{1}{(e^{\frac{1}{z}}-1)(e^z-1)}.$$ Trying to integrate this function in a closed contour around $0$ has been impossible to me. Let $\epsilon&...
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1answer
71 views

Contour integral around an essential singularity

The complex function $$f(z)=\frac{1}{e^{1/z}-1}$$ has an essential singularity at $z=0$, and an infinite quantity of poles inside every open neighborhood containing it. Let $\mathbb{R}\ni\epsilon>...
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Compute integral using Cauchy Principal Value

Using the Cauchy Principal Value, I need to compute the following integral $$\int_{-\infty}^\infty\frac{\cos(ax) - \cos(bx)}{x^2}dx$$ I have used the standard semi-circle contour with an indentation ...
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32 views

How do I solve the distributional equation T.x =1?

I am having a bit of trouble solving distributional equations. An example that I am currently working on is to show that the distributional equation $T.x = 1$ has a solution if and only if $T=p.v.(\...
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1answer
68 views

Calculate the integral v.p$\int_{-\infty}^{\infty} \frac{e^{px}}{1-e^{x}}\, dx$ [closed]

i need help with the following integral please: v.p $\int_{-\infty}^{\infty} \frac{e^{px}}{1-e^{x}}\, dx$ $0<p<1$ Where v.p denotes the main value. I have tried to do it through excercises ...
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What is a method for solving principal value integral of $\frac{1}{\pi}\int_{-B}^{B} \frac{x \sqrt{B^2-x^2}}{x-y}\mathrm{d} x$?

Question: I am trying to solve a principal value integral involving a square root. Using Mathematica I can get an answer but I would like to know a general approach to obtain them by hand. To be ...
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1answer
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Jordan Lemma not applying

so I was trying to evaluate the integral of {1/(x-ia) dx} from -infinity to +infinity and where a>0 Of course, Jordan's lemma doesn't apply (it's what I used before, this is the case where the ...
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2answers
175 views

How do you integrate $\int_{0}^\infty \frac{\log(x)^2}{(1-x^2)^2}$ using contour integration?

I have tried using the standard keyhole integral, and looking at$\ \log(x)^3 $, but because the poles lie on the real axis, when I expand the integrand $\ \frac{(\log(x) + 2\pi i)^3}{(1-x^2)^2} $ I ...
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64 views

The Cauchy principal value of the Riemann Zeta function

In many online sources, you can find $$ \zeta(s) \overset{C.P.}{=} \lim_{\epsilon \to 0} \left(\frac{\zeta(s+\epsilon)+\zeta(s-\epsilon)}{2}\right). $$ This seems quite logical, but I neither know how ...
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4answers
62 views

Does this double integral give 0 or does it diverge?

I was evaluating a double integral using the following iterated integral: $$\int_{-\infty}^{\infty} \int_{0}^{1} \frac{2xy}{x^2+1} dy dx$$ This simplifies down to: $$\lim_{\lambda \to \infty} \left[\...
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1answer
49 views

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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44 views

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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2answers
73 views

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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3answers
50 views

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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1answer
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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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24 views

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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1answer
78 views

How to calculate principal value of this complex integral?

$$I = \text{p. v.}\int_{-\infty}^{\infty}\frac{e^{\alpha x}}{e^{2x} - 1}dx$$ $$ 0<\Re(\alpha)<2$$ Use this contour (see image) $$\lim_{\epsilon\rightarrow 0, \\ R\rightarrow\infty}{\int_{-R+i\...
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Compute the $PV\int_0^{\frac{π}{3}}\frac{\cos (4x)}{\cos (3x)}dx$

Problem : Evaluate the closed form of : $PV\displaystyle\int_0^{\frac{π}{3}}\frac{\cos (4x)}{\cos (3x)}dx$ Wolfram alpha give me : $I=PV\displaystyle\int_0^{\frac{π}{3}}\frac{\cos (4x)}{\cos (3x)}...
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$\int_{0}^{\pi/2}{\frac{dx}{a+\sin^2(x)}}$ for $|a|>1$ [duplicate]

Evaluate: $$\int_{0}^{\pi/2}{\frac{dx}{a+\sin^2(x)}}$$ for $|a|>1$ Note: This integral is to be solved using complex analysis (Calculus of Residues)
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Differentiating an integral understood as Cauchy Principal Value

Let's say that the integral $$\int_a^x f(t)\,\mathrm{d}t$$ can only be understood as Cauchy principal value. So, the Riemann integral doesn't exist, and the improper integral diverges, but the Cauchy ...
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Principal value for a case with many variables involved

I'm having problems in an integral of many variables in which one of them leads to a Dirac delta and a Principal value. I have to solve \begin{equation} Int = \int dp dp' dk dk' dq f[p,p'] f[k,k'] ...
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Cauchy Principal value of $\int_a^b \frac{dt}{\pi}\frac{\sqrt{(t-a)(b-t)}}{x-t}t$

I am trying to determine the Cauchy Principal value of the following (with $b>a$, and $x\in (a, b)$) $\int_a^b \frac{dt}{\pi}\frac{\sqrt{(t-a)(b-t)}}{x-t}t$ I naively thought it to be $-2\pi ix\...
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41 views

Integral equation exercise

Let $\phi \in L^2\left(-\infty,\infty\right)$ and $|x|\geq a$. I have that the following integral equation for $\left(\partial\phi/\partial y\right)\left(z,0,t\right)$: $$\frac{\partial\phi}{\partial ...
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1answer
81 views

Prove that Cauchy principal value of $\int\limits_{\mathbb{R}}\frac{e^{-x^2}}{x+2}dx=\frac{π\text{erfi}{2}}{e^4}$

Prove that : $$PV\displaystyle\int_{\mathbb{R}}\frac{e^{-x^2}}{x+2}dx=\frac{π\text{erfi}{2}}{e^4}$$ I know that for $a \le p \le b$ $$PV\int_{[a,b]}f(x)dx =\lim_{t\to 0^+} \left(\int_{[a,c-p]}f(x)...
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1answer
72 views

Find Cauchy principal value of the following integral : $\int\limits_{\mathbb{R}}\frac{e^{-x^3}}{x+3}dx$

Find the principal value of : $\displaystyle\int_{\mathbb{R}}\frac{e^{-x^3}}{x+3}dx$ $\displaystyle\int_{\mathbb{R}}\frac{e^{-x^2}}{x+1}dx$ Of course wolfram doesn't say convergence because ...
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82 views

Reduction of a type of elliptic integral to Legendre normal form

Define the set $D$ to be the subset of $\mathbb{R}^{5}$ specified as follows: $$D:=\{\left(a,b,c,d,z\right)\in\mathbb{R}^{5}\mid c>0\land z\ge a\land z>d\}.$$ For each $\left(a,b,c,d,z\right)\...
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4answers
166 views

Compute an integral about error function $\int_{-\infty}^{\infty} \frac{e^{-k^2}}{1-k} \mathrm{d}k$

There is an integral $$ \mathcal{P}\int_{-\infty}^{\infty} \frac{e^{-k^2}}{1-k} \mathrm{d}k $$ where $\mathcal{P}$ means Cauchy principal value. Mathematica gives the result (as the screent shot ...
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Analytic continuation and complex integration of one variable of a multivariate function

Consider a function $f:(x_0,x_1,...,x_n)\in\mathbb{R}^n\rightarrow f(x_0,...,x_n)\in\mathbb{R}$. $f$ is a ratio of polynomials in $x_0,...,x_n$ which only has simple poles in the variable $x_0$, whose ...
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$\int_{-\infty}^\infty \frac{e^{pz}}{e^z-1}dz$ Cauchy principal value

$$\int_{-\infty}^\infty \frac{e^{pz}}{e^z-1}dz$$ I started by defining the following contour: rectangular contour It is easy to show that the integrals along the 2 vertical sides of the rectangle ...
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1answer
80 views

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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75 views

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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1answer
110 views

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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54 views

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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61 views

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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177 views

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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1answer
358 views

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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1answer
53 views

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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1answer
482 views

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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47 views

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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1answer
82 views

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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1answer
113 views

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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1answer
292 views

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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1answer
103 views

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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3answers
163 views

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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0answers
40 views

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. ...