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.

Filter by
Sorted by
Tagged with
-1 votes
1 answer
76 views

How to evaluate this by contour integral?

I want to evaluate cauchy principal value by complex contour integral, but I failed. Is this integral solved by contour integral? $$PV\int_0^\infty\frac{1}{x^3-1}\,\mathrm{d}x$$
ssam's user avatar
  • 3
2 votes
0 answers
56 views

Motivation behind Cauchy principal value of an improper integral? [duplicate]

According to the Wikipedia, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. The ...
FreeMind's user avatar
  • 2,539
0 votes
0 answers
13 views

Asymptotic behavior of a 2D integral: retarded/advanced green's function with saddle point dispersions

I want to analytically evaluate the 2D integral for some real $E< 1$ $I^\pm(E)=\lim_{\eta \rightarrow 0}\int_{-1}^{1} dx dy \frac{1}{E\pm i\eta -xy}$ In particular, I want to understand the ...
Yidan's user avatar
  • 65
2 votes
0 answers
26 views

What is the relationship between the two versions of the Sokotski-Plemelj theorem?

Wikipedia gives the follow statement of the general Sokhotski-Plemelj theorem: Let $C$ be a smooth closed simple curve in the plane, and $\varphi$ a complex-analytic function on $C$. Define $$ \phi_i(...
tparker's user avatar
  • 6,189
1 vote
1 answer
63 views

Confusion in working out $\mathcal{P} \int_{-\infty}^{\infty} \frac{dz}{z-i}$.

I'm trying to work out $$I=\mathcal{P} \int_{-\infty}^{\infty} \frac{dz}{z-i}$$ where $\mathcal{P}$ denotes the fact we are taking the Cauchy principal value. Let $\gamma$ be the contour such that $\...
Robin's user avatar
  • 3,185
1 vote
1 answer
57 views

Hilbert transform of the integral of a function

Given that f and g are Hilbert transform pair $$Hf(x) = g(x)$$ Although the derivative will maintain the transform pair relation $$Hf'(x) = g'(x)$$ Does the Hilbert transform pair relation apply to ...
A AlOmar's user avatar
1 vote
0 answers
65 views

Using quantile function of eventually monotonic pdf to get something like an expectation

Consider a random variable on $\mathbb{R}$ that may or may not have an expectation. (E.g., if its pdf is a Cauchy distribution, it won't.) Let $p(u)$ be a probability density function on $\mathbb{R}$. ...
HW.'s user avatar
  • 53
2 votes
1 answer
86 views

Can the inverse Fourier transform $f(x)=\mathcal{F}_{\omega}^{-1}[F(\omega)](x)$ converge when $F(\omega)=\mathcal{F}_x[f(x)](\omega)$ doesn't?

Are there examples of functions $f(x)$ for which the inverse Fourier transform integral $$f(x)=\mathcal{F}_{\omega}^{-1}[F(\omega)](x)=\int\limits_{-\infty}^\infty F(\omega)\, e^{2 \pi i x \omega}\, d\...
Steven Clark's user avatar
  • 7,290
1 vote
0 answers
71 views

Finding the Cauchy Principal Value of $\int_{-\infty}^{\infty}\frac{1}{x-i}dx$

I know that the Cauchy Principal value of $\int_{-\infty}^{\infty}\frac{1}{x-i}dx=i\pi$ but I do not understand how to get to this. I understand how to use the principal value for functions with real ...
ninanain's user avatar
4 votes
1 answer
98 views

Cauchy Residue Theorem with Logarithms

I've been trying to use Cauchy's Residue Theorem to calculate the following integral, but I am getting stuck with the logarithm and not sure how to proceed. Q: Evaluate $\int_{-\infty}^{\infty} \frac{...
user38372's user avatar
0 votes
1 answer
143 views

Numerical Principal Value integration

I want to calculate the Principal Value Integral of $$ \int_a^\infty dx \frac{f(x)}{(x-b)^n}, $$ where $b$ is in the integration region. For the case $ n=1 $ I do know the trick $$ \int_a^\infty dx\...
Nik's user avatar
  • 1
0 votes
3 answers
126 views

How to evaluate real and imaginary part of $\Gamma\left(\frac{2}{3},-\frac{1}{3}\right)$

I calculated the principal value of the following integral: $$PV\int_{0}^{\infty}\frac{t^{\frac{1}{3}}e^{-t}}{1-3t}dt=\left(\frac{\Im\left[\left(-1\right)^{\frac{5}{6}}\Gamma\left(\frac{2}{3},-\frac{1}...
Math Attack's user avatar
2 votes
2 answers
44 views

Decomposing a denominator in terms of principal value

I am trying to understand a step made in a physics textbook ( Lectures on Quantum Field Theory, by Ashok Das ). But I don't even know what the formula is called, or what are the keywords. So I couldn'...
baba26's user avatar
  • 147
0 votes
0 answers
40 views

Finding Cauchy principal value of contour integral

Consider a triangle on the complex plane, these three points are $z=1,\omega,\bar{\omega}$, here $\omega=e^{\frac{2}{3}\pi i}$. The boundary of triangle is $\partial T$, whose direction is ...
Cunyi Nan's user avatar
  • 698
2 votes
3 answers
149 views

Principal value of 1/|x|

Can we define the Cauchy principal value of $\frac{1}{|x|}$? If yes, how does it act on a smooth function with a compact support? Is it defined as $$\text{p.v.} \frac{1}{|x|}(\psi) = \lim_{\epsilon \...
Cohen Lu's user avatar
  • 283
0 votes
2 answers
111 views

How to calculate $\int_{-\infty}^{\infty}\frac{\cos^{3}x}{x^2+1}\mathrm{d}x$ with the help of complex function?

I have met this improper integral: $$\int_{-\infty}^{\infty}\frac{\cos^{3}x}{x^2+1}\mathrm{d}x$$ I tried to use residues but it doesn't work. The singularity point of function $f(z)=\frac{\cos^{3}z}{1+...
Cunyi Nan's user avatar
  • 698
3 votes
1 answer
123 views

Second distributional derivative of $P.V. \frac{1}{x}$

I computed first derivation and I get that $$\langle(\mathcal{P}\frac{1}{x})', \varphi\rangle = v.p. \int_{ \mathbb{R}} \frac{\varphi(0) - \varphi(x)}{x^2} dx$$ In order to get second derivative we ...
Margaret's user avatar
  • 107
0 votes
0 answers
16 views

Optimization of numerical evaluation of Cauchy principal value

I need an efficient way to evaluate the Cauchy principal value of a function so that I can do it thousands of times in a timely manner. It boils down to this: $g(t,x,y)=P\int_0^\infty \frac{f(u,x,y)}{...
Bidon's user avatar
  • 393
6 votes
1 answer
311 views

Finite part of $-1/x^2$

I'm learning the basic of Distributional Theory. I ended up solving the following exercise: 'Find the distributional derivative of $P.V.1/x$'. After few computation, I arrived at the following: $$\...
Nick's user avatar
  • 85
0 votes
0 answers
41 views

Stieltjes transformation of an even measure

I am confused about the behavior of the Stieltjes transform $s_\mu(x)$ for an even measure $d\mu(x)$. Let $d\mu(x)$ be positive and even on $[-1,1]$; the Stieltjes transform is the function $$ s_\mu(z)...
hulsey's user avatar
  • 151
1 vote
0 answers
55 views

Kramers-Kronig computation for real susceptibility

i am trying to get the real part of electric susceptibility using the imaginary part with Kramers-Kronig relation for a Lorentz-Drude model.I chose to ask this question in math stack exchange as im ...
zero's user avatar
  • 11
2 votes
0 answers
35 views

Analytic continuation of integral representation as Cauchy principal value

Suppose I have the following formula, \begin{equation} \mathrm{P}\int_{-\infty}^{\infty} f(x,q) dx = F(q), \end{equation} for all $q\in\mathbb{R}$, where P stands for the Cauchy principal value. If $F(...
norio's user avatar
  • 290
1 vote
0 answers
29 views

Are there arguments for not using Cauchy Principal Value when there's odd singularity?

Question: Are there arguments or examples that shows I should not always use Cauchy Principal Value when Riemann's integral is not defined for mathematical applications? I take the case which $f(x)$ ...
Carlos Adir's user avatar
  • 1,292
1 vote
1 answer
87 views

Principal value integral on half-interval

Is there a way to define the integral of say $f(x)=-1/x^2$ between 0 and y so that it is safely valued as $F(y)=1/y$? Put another way, usually the Cauchy integral $F(x)$ is defined symmetrically ...
Douar Gwenn's user avatar
1 vote
2 answers
99 views

How to prove $P \int_{-\infty}^\infty \frac{e^{-ax^2-bx}}{x} dx = -i\pi$?

For $a,b>0$, Mathematica says that $$P \int_{-\infty}^\infty \frac{e^{-ax^2-bx}}{x} dx = -i\pi,$$ where $P$ denotes the principal-value integral. How can I derive the above? Due to the factor $e^{-...
Laplacian's user avatar
  • 2,484
2 votes
1 answer
71 views

$\lim_{\epsilon \rightarrow 0^+} \int_{-1}^1 \frac{f(t)}{t-i\epsilon}dt = \text{PV} \int_{-1}^1 \frac{f(t)}{t} dt + i\pi f(0)$

A textbook I am going through claims that for smooth $f: \mathbb{R} \rightarrow \mathbb{R}$, it holds that $$\lim_{\epsilon \rightarrow 0^+} \int_{-1}^1 \frac{f(t)}{t-i\epsilon}dt = \text{PV} \int_{-1}...
CBBAM's user avatar
  • 5,637
3 votes
1 answer
221 views

Continuity of Hilbert transform

Suppose $f : \mathbb{R} \to \mathbb{R}$, be a non-negative, bounded and continuous function, and its support is a compact interval in $\mathbb{R}$. Moreover, we have that $\int f(x) \, dx =1$. The ...
Abdullah123's user avatar
1 vote
0 answers
43 views

Can you convolve two generelized functions? how?

By generalized functions I mean functionals from $C^\infty_c(U)\to \mathbb{R}$ as defined in wikipedia. I want to compute: $$ \operatorname{comb}(\omega)*\operatorname{pv}(\frac{1}{\omega}) $$ But the ...
Alexey's user avatar
  • 596
2 votes
2 answers
327 views

Cauchy principal value: methods

I don't understand yet the logic in the Cauchy Principale Value (P.V.) calculations. Let the resideu theorem: $$\color{red}{\oint_Cf(z) \ dz = 2\pi i \sum_{k=1}^n \underset{z=z_k}{Res}\{f(z)\}}$$ (we ...
user avatar
0 votes
1 answer
83 views

Average value of complex exponential

I'm interested in evaluating the following limit $$\lim_{T\to\infty}\frac{1}{T}\int_0^T dt\;e^{i(\varepsilon'-\varepsilon)t},$$ with $\varepsilon$ and $\varepsilon'$ being real numbers. Performing the ...
11Elves's user avatar
  • 15
0 votes
1 answer
186 views

Contour Integral with Multiple Poles

I'm reading through a physics textbook and came across the integral in the image below. The author says we're using contour integration for this, but I'm not sure how they are getting 2 terms in the ...
stable_pendulum's user avatar
0 votes
1 answer
43 views

Some clarification about this integration result

I am studying special function, in particular the Exponential Integral, and I came up with this integral: $$\int_{-3}^3 \frac{e^{i ax}}{x}\ \text{d}x$$ Now, I understood that this integral does not ...
Martin and Friends's user avatar
4 votes
0 answers
117 views

Convergence of Hilbert transform of a converging sequence

Fix $n$, and consider random variables $x_1, \dots, x_n$ whose joint p.d.f. is $p_n$. Assume that the empirical distribution of $x_1, \dots,x_n$ converges weakly almost surely to the probability ...
Rostam22's user avatar
  • 472
3 votes
1 answer
110 views

An "asymmetric" Cauchy principal value distribution

I'm familiar with the usual Cauchy principal value distribution: $$ PV[\phi(x)] := \lim_{\epsilon\to 0^+}\int_{\mathbb{R}\setminus[-\epsilon,\epsilon]}{\dfrac{\phi(x)}{x}}\,dx = \int_{0}^{\infty}{\...
Patch's user avatar
  • 4,203
1 vote
2 answers
195 views

Keyhole contour for integral with pole on cut

$$\textrm{p.v.}\int\limits_0^\infty \frac{\ln x \ dx}{x^4-16}$$ I have to solve this integral. So I got an idea I've seen several times, here it is: let's take a keyhole contour with cut on positive ...
Big Coconut's user avatar
1 vote
1 answer
60 views

Solve an integral equation in which there is Cauchy-type singularity

I'm looking for a solution to an integral equation: Given a function $f(x)$, find a function $g(x)$ such that $$\int_0^1 \frac{g(s)}{x-s} ds = f(x), \forall x \in [0,1]$$ satisfies. Function $f(x)$ is ...
swang's user avatar
  • 13
0 votes
2 answers
96 views

Difficulty finding the principal value of $\int_{-\infty}^{\infty}\frac{\cos3x}{x-1}\,\mathrm{d}x$

I am having problems in evaluating the following Cauchy principal value $$\int_{-\infty}^{+\infty} \frac{\cos(3x)}{x-1}d x$$ I know it is supposed to show my work, but I got barely nothing. I tried ...
Martin and Friends's user avatar
1 vote
0 answers
29 views

Legitimate or illegitimate manipulations within a principal value integral

First of all, let me excuse myself by saying that I'm a physicist, so I tend to manipulate things a bit thoughtlessly and hope for the best. Still, while manipulating some expressions today I found ...
Alex V.'s user avatar
  • 283
1 vote
0 answers
61 views

Meaning of the integral $I(t)=\lim_{\epsilon\rightarrow 0^+} \int_0^t\frac{\phi(y)}{(\epsilon +iy)^n}dy$

I am trying to give a meaning to this integral $$I(t)=\lim_{\epsilon\rightarrow 0^+} \int_0^t\frac{\phi(y)}{(\epsilon +iy)^n}\,dy$$ where, $n > 1$, $\phi(y)$ is a complex function, infinitely ...
quantum.stuck's user avatar
2 votes
1 answer
58 views

Computing the asymptotics of a principal value integral

I have been looking at the following principal value integral (with $x>0$ and $0< \sigma < 1$) $$ \mathrm{P.V.} \int_0^\infty \frac{v^{-\sigma} e^{-x v}}{1 - v} dv, $$ and I would like to ...
qgp07's user avatar
  • 185
1 vote
2 answers
166 views

Сalculate the integral PV $\int_{0}^{\infty} \frac{dx}{x^\alpha(x-a)}dx.$

Сalculate the integral $$ \mathrm{PV}\hspace{-0.5ex}\int_{0}^{\infty} \frac{dx}{x^\alpha(x-a)}, $$ where $0<\alpha <1$ and $a>0$. So we have simple poles $z = 0$ and $z = a$; We can build ...
rpr's user avatar
  • 55
0 votes
1 answer
68 views

Evaluate $PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx. a>0$

Evaluate $PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx. a>0$ using residues. So I have a theory how to calculate $PV \int_{-\infty}^{\infty} f(x)e^{iax}dx$ a>0, but I don’t know how to ...
rpr's user avatar
  • 55
3 votes
1 answer
271 views

Why do we use exponentials while integrating trigonometric functions in complex analysis

Let p(x) be some polynomial function. Now, we have an integral of the form : $$I=\int_{-\infty}^{\infty} \frac{\cos(x)}{p(x)}dx$$ What is usually done is that, we define this integral as : $$I'=\int_{-...
Nakshatra Gangopadhay's user avatar
2 votes
0 answers
39 views

Why is this estimator with ill-defined moments useful? And why is the Cauchy PV of its expectation integral a reasonable measure of center?

This question pertains to the paper (available online through JSTOR): M. H. QUENOUILLE, NOTES ON BIAS IN ESTIMATION, Biometrika, Volume 43, Issue 3-4, December 1956, Pages 353–360, https://doi.org/10....
Aaron Hendrickson's user avatar
6 votes
1 answer
98 views

Why does the Cauchy PV of $\mathsf E(1/X)$, $X\sim\mathcal N(\mu,\sigma^2)$ accurately reflect the sample mean when $|\sigma/\mu|$ is small?

Suppose $X\sim\mathcal N(\mu,\sigma^2)$. The first negative moment $\mathsf E(1/X)$ does not exist; however, we can define it in the sense of the Cauchy principal value: $$ \tag{1} \mathsf E(1/X)\...
Aaron Hendrickson's user avatar
2 votes
1 answer
195 views

Finding $\text{PV}\int_0^\infty\frac{\sec\left(\pi B(xt-\lfloor xt+\frac12 \rfloor\right)-\sec\left(\pi B(x-\lfloor x+\frac12 \rfloor\right)}{x}dx$

The following problem is proposed by a friend $$\text{PV}\int_0^\infty\frac{\sec\left(\pi B(xt-\lfloor xt+\frac12 \rfloor\right)-\sec\left(\pi B(x-\lfloor x+\frac12 \rfloor\right)}{x}\mathrm{d}x,\quad ...
Ali Shadhar's user avatar
  • 25.2k
1 vote
1 answer
186 views

Cauchy principal in the definition of fractional laplacian

Let $\alpha \in \mathbb{R}$,$\ 0<\alpha <2 \ $, the fractional laplacian is defined as: \begin{equation}\label{def} (-\Delta)^{\frac{\alpha}{2}} u (x) := C_{n,\alpha} \lim\limits_{\epsilon \...
Annabelle's user avatar
3 votes
0 answers
69 views

Generalization of central limit theorem to sums of the form $\sum_{k=1}^n\frac{1}{X_k^m}$, $m=1,2,\dots$?

Consider this lesser-known fact: If $X\sim\mathcal N(\mu,\sigma^2)$ then $$ \frac{\frac{1}{n}\sum_{k=1}^n\frac{1}{X_k}-\mathsf E_\mathcal PX^{-1}}{\pi f_X(0)}\overset{d}{\to}\operatorname{Cauchy}(0,1)...
Aaron Hendrickson's user avatar
12 votes
1 answer
387 views

What exactly do delta method estimates of moments for $1/\bar X_n$, $\bar X_n\sim\mathcal N(\mu,\sigma^2/n)$ approximate? (not as simple as you think)

Let me start with the excerpt out of Casella & Berger's Statistical Inference (2nd edition, pg. 470) that inspired this question. Definition 10.1.7 For an estimator $T_n$, if $\lim_{n\to\infty}...
Aaron Hendrickson's user avatar
1 vote
0 answers
34 views

Is $\int dx dy \frac{f(x) g(y)}{y-x + i0}$ defined on piecewise continous functions

Question: Let $f,g$ be piecewise continous square-integrable functions. Then, is the following limit integral well-defined (and finite)? $$ \lim_{\epsilon\to 0} \int dx dy ~ \frac{f(x) g(y)}{y-x + i\...
Cream's user avatar
  • 392

1
2 3 4 5 6