Questions tagged [cauchy-principal-value]

Question 1

EDIT: Solved, stupid mistake when doing the cube of the binomial... I am having some trouble with principal value integrals. From what I understood I should get a $\pm \pi i Res(f,z_o)$ contribute ...

Question 2

Let $$F(z)=\int_{-1}^1 \sqrt{\frac{1+t}{1-t}}\frac{dt}{z-t},$$ where $\text{Im}(z)>0$. I know that result is $F(z)=-i\pi\sqrt{\frac{1+z}{1-z}}-\pi$, but don't know how to prove it. In Szego book &...

Question 3

I have the following integral: $$I= \lim_{\epsilon\rightarrow 0}\int_{-\infty}^{+\infty} dw \frac{e^{-iA(w-\tau)^2}}{w+i\epsilon}$$ where A and $\tau$ are constants. I tried using the Sokhotski-...

Question 4

I am trying to evaluate the following integral $$ \oint_C z^k \log|1-z|dz $$ where $C$ is the complex unit circle. Attempt 1 I have attempted the approach used in this answer where we take the ...

Question 5

Since the Gamma function has a pole on 0 and all negative integers, with the signs alternating, out of curiosity I want to investigate the area under gamma in the negative half-plane with all the ...

Question 6

The following problem was proposed online: $$\operatorname{P.V.}\int _0^1\frac{\ln ^2\left(x\right)}{1-2x}\:\mathrm{d}x=i\pi \ln ^2\left(2\right)+\operatorname{Li}_3\left(2\right)$$ $\displaystyle\...

Question 7

I integral the elliptic function in complex plane when $1<Re[z]\le k^{-1}$: $F(z)=\int_0^z\frac{du}{\sqrt{1-k^2u^2}\sqrt{1-u^2}}(0<k<1)$ Since the integrand has a singularity at (u = 1), we ...

Question 8

I'm wondering if there is a way to define a distribution with the principal value of $$\frac{1}{r^3}=\frac{1}{(x^2+y^2+z^2)^{3/2}}$$ in analogy in $\mathbb{R}$ with the principal part of $\frac{1}{...

Question 9

I'm starting my masters in theoretical condensed matter physics soon, and I encountered this integral in a paper I have to read before starting my research. The paper presented the answer as: \begin{...

Question 10

I want to evaluate the Cauchy principal value of, $$\int_{0}^{2}\frac{dx}{\sqrt {x}\ln(x)}$$ As per the definition of Principal value, $$\text{PV}\int_{a}^{b}f(x)dx=\lim_{q\to0^{+}}\int_{a}^{c-q}f(x)...

Question 11

In the context of https://arxiv.org/pdf/hep-ph/9805477, one finds $\mathcal{P}\int_{-1}^1dx \frac{N(x)}{(x-x_0)\sqrt{1-x^2}}$ where N is an anaytic function of x. I care about the case with $-1<x_0&...

Question 12

I need help to evaluate this integral : $$\Omega = p.v \int^{\infty}_{0} \left({p.v \int^{\infty}_{0} \frac{\text{Li}_n\left({\frac{y}{\sinh(y)}}\right)dy}{B^2 x^2-y^2}}\right)dx$$ B:Constant $$\text{...

Question 13

I wish to compute the principal value integral, \begin{align} I = PV \int_{-u}^{+\infty} \frac{e^{-(u+s)^2}e^{-i\Omega s}}{s^2} ds \end{align} Splitting the integrand up, we get, \begin{align} I &=...

Question 14

Let's consider: $$\int_{\frac{\pi }{2}}^{\frac{3 \pi }{2}} \tan (x) \, dx$$ The graph of the function within the integration interval looks like this: Sorry if the answers are obvious, but my ...

Question 15

The Sokhotski-Plemelj Formula states \begin{equation} \frac{1}{x \pm \textbf i \eta} = \mathcal{P} \left(\frac1x\right) \mp \textbf i\pi \delta(x) \end{equation} where this expression has to be ...

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Questions tagged [cauchy-principal-value]

calculate Principal Value of $\frac{x}{x^3-1}$ (residue th)

Cauchy type integral of Chebyshev weight of third kind $F(z)=\int_{-1}^1 \sqrt{\frac{1+t}{1-t}}\frac{dt}{z-t}$

Evaluating the principal value integral $PV\int_{-\infty}^{+\infty} dw \frac{e^{-iA(w-\tau)^2}}{w} -i\pi e^{-iA(\tau)^2}$

Contour integral of $z^k \log|1-z|$ around the unit circle

Principal value integral of the negative side of the gamma function

Issues when calculating the Cauchy principal value of $\int _0^1\frac{\ln ^2\left(x\right)}{1-2x}\:\mathrm{d}x$

How to compure Elliptic function using complex integral and Cauchy principal value

3D extension of the Principal part 1/x distribution

(Still) Stuck on an integral of the form: $\mathcal P \int_0^\infty \frac{x \tanh x}{a^2-x^2} \mathrm dx$

How do I evaluate the Cauchy principal value of $\int_{0}^{2}\frac{dx}{\sqrt {x}\ln(x)}$?

Integrals of the form $\mathcal{P}\int_{-1}^1dx \frac{N(x)}{(x-x_0)\sqrt{1-x^2}}$

Evaluate $ p.v \int^{\infty}_{0} \left({p.v \int^{\infty}_{0} \frac{\text{Li}_n\left({\frac{y}{\sinh(y)}}\right)dy}{B^2 x^2-y^2}}\right)dx$ [closed]

Principal value integral of $f(x)/x^2$

Question about $\int_{\frac{\pi }{2}}^{\frac{3 \pi }{2}} \tan (x) \, dx$

Does a generalization of the Sokhotski-Plemelj Formula to four (or higher) dimensions exist?

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