Cauchy Principal Value

Cauchy principal value integral example. You learn in calculus courses that an improper integral is sometimes divergent, but in this video I show you how to make it (rigorously) equal to zero! This is widely used in distribution theory and Fourier analysis

[johetajava]

Thanks for the video, you are an amazing teacher! I found this because my task's answer sheet said the function diverges, and I messed up and during the process, and actually calculated the Cauchy principal value. The video cleared up everything. :)

[RalphDratman]

i like this very much because it is conceptual, not just algebraic. Beneath every demonstration by algebra there must be a concept, wouldn't you agree? Otherwise, how do you know what steps to take in your algebra?

[ozzyfromspace]

I assumed this was possible when I was little (10th grade, way back then) but I didn't have the language of "principle value", so this video brought a lot of things full circle. Thanks PiM ☺️🙌🏽😁

[michalbotor]

dr peyam,
could you make a video or two about calculating riemann integral from definition?

[_DD_]

Ehm.. let's be honest, if it wasn't for all the mathematical theorems... 1/x is perfectly symmetric, therefore conceptually it does make sense stating that 2 symmetric equal parts cancel each other if they are of opposite sign. Now you can argue about infinity not being a number therefore the concept of subtracting infinite areas between each other might not have sense. But, they both go to infinity with the same speed and the same law, so conceptually it's not wrong

[happyfrogkhmel]

Thank you so much for your job! You are a wonderful teacher)

[Tengdbuddy]

The Fourier Transform of the Heavyside function please. <3 So much love for your work! <3

[MadhavBhardwajMln]

So whats the conclusion ? Does it diverge or converge ? Or it depends on the way we define the integral?

[utuberaj]

OMG- Dr Peyam- you opened my eyes about this "new" concept about the PV- which I heard it today while watching another video(@ Owl Math) who name dropped "Cauchy Principal Value" in his recent video on this discontinuous integrand.
What was really nice was the very simple way you explained the change of the limits to a common variable \epsilon for both the integral pieces- which is what Calc-2 text books teach you and the answer is -DIVERGENT- though this is very counter-intuitive- when you see the graph.
But this begs the question: Are we to use the result from the Fundamental Theorem Of Calculus - OR the fid the PV?
Could you make another follow up video on this and show us examples using the Dirac delta function or the Heaviside Function- and any others for me to learn mor about this beautiful concept please.
I love your videos as you explain complex stuff with child-like enthusiasm, and lots of LoL too. Amazing Prof. (reminds me of Feynman!!)

[helloitsme]

I can't grasp why people say this integral is undefined, I mean an integral is a limit itself already and the integral limits 0 so why not?

[amanda]

tysm! really helped me understanding bout CPV better. stay safe and healthy :)

[studentofspacetime]

The problem with this approach, is that you chose the interval to be [-eps, +eps], because you had a symmetric situation to begin with. You could have also chose [-eps, 2*eps] for instance, and then the limit would diverge.
What if you had a random function that is neither even nor odd?

[nullplan]

I thought inf - inf is an indeterminate form, isn't it? So a closer analysis is required to find the actual value.

[rmd]

very instructive video. This topic was one of the hardest concepts I met in my studies when I was young...A little question: when you write PV[(1/x)]U, you mean what sometimes we find written as <PV(1/x), U>, so the definition of generalized function, right?

[wahyuhidayat]

I like watching your video....it's awesome

[matteo]

Is it possible to show that taking the two separate limits and the single one is just the same thing?
Is it possible show that by splitting the integral in [s, 1] into [s, -t] [-t, 1] only the one in [-t, 1] isn't zero?

[tgx]

Please, when I studied, we learned, when the area above the axis x is infinity & area below the axis is infinity, then Lebesgue integral doesn't exist.If Riemann integral will be zero, on the internet is written, if exists Riemann integral then exists Lebesgue integral.Can I say that using Cauchy teorem is L integral=0??According standard definition L integral dosn't exist.If integrals L, R, N do not exist,  what type (name) of integral =0? Or is this integral without the name, only as area?It only makes sense. Thank you.

[matejvanek]

Hi, I have no idea how ln(-1) can equal 0? Isnt it undefined?

[Bombelus]

Can't I just say it's zero because f(x) = 1/x is odd?

[TheGalactik]

So guuud. I love mathematics whenever I'm in a university of telecommunication and Network :(