Talk:Cauchy principal value

Notation

I would rewrite the article, using notation

$-\!\!\!\!\!\!\int _{a}^{c}f(x){\rm {d}}x$

In the current version, the "definitions" are a little but fussy. dima (talk) 02:29, 28 July 2008 (UTC)

I would only use this notation if you think it is worth introducing to readers who have never seen it before. The notation is not common, from what I can tell.Njerseyguy (talk) 07:53, 11 March 2009 (UTC)

Confusing sentence in "Formulation"

"In the case of Lebesgue-integrable functions, that is, functions which are integrable in absolute value, these definitions coincide with the standard definition of the integral."

I don´t understand the intened meaning of this sentence, since Lebesgue-integrable functions ≠ functions which are integrable in absolute value. The class of Lebesgue-integrable functions is broader than that.--79.235.151.134 (talk) 09:56, 20 April 2010 (UTC)

Almost the only?

"It is the inverse distribution of function x and is almost the only distribution with this property" What does this mean? 78.91.83.39 (talk) 16:51, 22 May 2013 (UTC)

Definition

I think the requirement that the lateral limits exist as extended reals is not necessary. Considere the example:

$f(x)={\frac {1}{x^{2}}}\sin \left({\frac {1}{x}}\right)$

We see that

$\int _{\varepsilon }^{a}f(x)dx=\cos \left({\frac {1}{a}}\right)-\cos \left({\frac {1}{\varepsilon }}\right),~~\forall \varepsilon >0$

which implies the lateral limit does not exist. Nonetheless, I would say that

$-\!\!\!\!\!\!\int _{a}^{b}f(x)\,\mathrm {d} x=\lim _{\varepsilon \rightarrow 0+}\left[\int _{a}^{-\varepsilon }f(x)\,\mathrm {d} x+\int _{\varepsilon }^{b}f(x)\,\mathrm {d} x\right]=\int _{-a}^{b}f(x)dx=\cos \left({\frac {1}{b}}\right)-\cos \left({\frac {1}{a}}\right),~~a<0<b$

where I have used that f is odd so that

$\int _{a}^{-\varepsilon }f(x)\,\mathrm {d} x+\int _{\varepsilon }^{-a}f(x)\,\mathrm {d} x=0,~\forall a<0.$

By the way, I see no problem in saying that

$-\!\!\!\!\!\!\int _{a}^{b}\mathrm {d} x=b-a$.

Do you agree with me? Lechatjaune (talk) 17:27, 26 June 2014 (UTC)

Infinite number

I don't understand the meaning of the heading in Formulation under point 2: 'infinite number'.Madyno (talk) 16:36, 2 January 2018 (UTC)

Presumably it's talking about a singularity at infinity rather than at some number b. This should probably be clarified, but then again, this article has a lot of other problems too. –Deacon Vorbis (carbon • videos) 18:12, 2 January 2018 (UTC)

Madyno, I had exactly the same question when I came across the article just now. I made a few changes to try to rectify this, but I also agree with Deacon Vorbis that more needs to be done. Csmallw (talk) 02:06, 7 January 2018 (UTC)

The Sokhotski–Plemelj theorem

Currently the article says "If the function $f(z)$ is meromorphic, the Sokhotski–Plemelj theorem relates the principal value of the integral over C with the mean-value of the integrals with the contour displaced slightly above and below, so that the residue theorem can be applied to those integrals."

According to the article on the Sokhotski–Plemelj theorem, it seems that this is only true if the singularity is a simple pole. For instance, the function $1/z^{2}$ has no principal value integrated on the real interval $[-1,1]$ since the limit diverges, but it does have displaced contour integrals above and below, which are in fact equal, since it has an antiderivative $-1/z$ on $\mathbb {C} \setminus \{0\}$. Is that right? Michael Shulman (talk) 19:35, 4 March 2024 (UTC)