*An exploration of the various methods of evaluating the improper integral of sin(x)/x* 0:00 The Feynman Technique 11:30 - The Method of Laplace Transforms 16:33 - The Method of Complex Analysis 25:20 - Definite integral series decomposition
I have 2 other takes for you : 1st : write sin(x) as its series representation and let u=x^2 , The goal is to obtain an integral of the form int [0,+oo] x^(s-1) * sum (-x)^n /n! * f(n) And to apply ramanujan’s master theorem that states that a integral of this form is equal to Gamma(s)*f(-s) 2nd : int[0,+oo] f*g = int[0,+oo] L(f)L^-1(g) , and you get your result immediatly L being the laplace transform
I have 2 more different way : 1.Use Mellin transform of sinx 2.another way of Feynman technique : sinx=Im(e^ix), let int(e^αx/x)=f(α), then f’(α)=int(e^αx)=1/α implies f(α)=lnα+c, so the integral=Im(ln(i)+c)=π/2😃👍
@ 21:05 You could have found the value of the integral over the bigger semicircle in the limiting procedure by using the same parametrization as the one for the smaller semicircle, but with R instead of epsilon.
At 25:10, since you got iπ = int [0,inf[ of e^ix/x, why isn't it allowed take the real part and get 0 = int[0,inf[ of cosx/x? I know it doesn't converge but what prevents us from doing that so?
I would personally recommend Princeton Lectures in Complex analysis by Stein & Shakarchi and Visual Complex analysis by Tristan Needham. Real analysis is an important course before Complex analysis as many of the concepts and proofs follow from it.
@@spiderjerusalem4009 For set theory I would recommend Jech's introduction to set theory. For real analysis I recommend Rudin's Principles of Analysis if you want a thorough book. If you want an introductory book I recommend Stephen Abbott's Understanding Analysis.
How can we know where does a contour integration work?, like, does it need a special structure?, or anything like a series of conditions?, Im trying to lean it trough your videos and that would help a lot, thanks c:
I have the following series would you be willing to take a look at it? series((digamma(k+3/4)-digamma(k+1/4))/((4k+2)(4k+3))) sum from k= 0 to inf This should have a closed form I believe, but I’m stumped on how one sums over digamma functions. You have one of the few channels with anything digamma related, would you be able to help?
Fascinating I'll have a look at it in the morning. Just to be sure I don't forget it, send it to via the email given in the about section of the channel.
Email sent, I should also say that the email is related to a problem you’ve tackled on your channel before, I posted the link to that problem in the email.
@@maths_505 Thank you. Is it impossible to do this purely by contour integration without any prior application of trig substitution or Feynman's technique?
I need your help❤ I'm syrian and im gonna graduate from high school soon and i have a good chance to study in the UK but I don't know what I should study Im a person who love maths ,physics and machines I love to study a sector of engineering but I dont have a final decision I want u to advice me and consider me your small brother 😂 I'm sorry this comment has no thing to do with ur video but I really need help note: I don't want to study maths in the university and thank u❤
Are you aware of this 1909 article by GH Hardy in the Mathematical Gazette, a magazine for mathematics teachers? He gives a number of methods, giving marks to each. math.harvard.edu/~ctm/home/text/others/hardy/sinx/sinx.pdf
*An exploration of the various methods of evaluating the improper integral of sin(x)/x*
0:00 The Feynman Technique
11:30 - The Method of Laplace Transforms
16:33 - The Method of Complex Analysis
25:20 - Definite integral series decomposition
Thanks bro
Could you add one more at the end for Lobachevsky's formula
@@maths_505 it seems that he can't
29:10 Lobachevsky’s integral formula
Thanks for putting all these methods together in one place. It really is a nice exercise!
YEEEAHHH.
We need more of these “every way to integrate…[insert interesting function here] “
Very instructive
the identity function.
A beautiful proof also for ∫ ₋ₒₒ⁺°° (¹/ₓ) ( sin x ) dx = π can be done using the Fourier transform.
The best part is the integral symbol seems like a snake😂🔥🔥
Was thinking about making this video 😂 AMAZING!
16:32 HE DID IT, HE SAID THE MAGIC WORD! (complex analysis)
Wonderful survey for this important integral. Thanks a lot.
Oh my god that was really toxic integral , i can't handle it. Nice 👍 keep goin
I have 2 other takes for you :
1st : write sin(x) as its series representation and let u=x^2 ,
The goal is to obtain an integral of the form int [0,+oo] x^(s-1) * sum (-x)^n /n! * f(n)
And to apply ramanujan’s master theorem that states that a integral of this form is equal to Gamma(s)*f(-s)
2nd : int[0,+oo] f*g = int[0,+oo] L(f)L^-1(g) , and you get your result immediatly
L being the laplace transform
What is the name of the formula used in your 2nd method?I have never seen it in the engineering mathematics course before.How to prove it?
For the complex analysis part?
That's Cauchy's residue theorem. You can find a proof in pretty much any textbook on complex analysis.
@@maths_505 thanks a lot
I love trigonometry, and this integral is just awesome :)
A real delight, thank you
I have 2 more different way :
1.Use Mellin transform of sinx
2.another way of Feynman technique : sinx=Im(e^ix), let int(e^αx/x)=f(α), then f’(α)=int(e^αx)=1/α implies f(α)=lnα+c, so the integral=Im(ln(i)+c)=π/2😃👍
@ 21:05 You could have found the value of the integral over the bigger semicircle in the limiting procedure by using the same parametrization as the one for the smaller semicircle, but with R instead of epsilon.
Been there done that
This time I just couldn't pass a chance to invoke Jordan boi's lemma
Would’ve been nice to see your reason for choosing a semicircular contour
At 25:10, since you got iπ = int [0,inf[ of e^ix/x, why isn't it allowed take the real part and get 0 = int[0,inf[ of cosx/x? I know it doesn't converge but what prevents us from doing that so?
The principle value is zero and that's all we can say about it
what book would you recommend for complex analysis? Would real analysis be vital prerequisite for it?
I would personally recommend Princeton Lectures in Complex analysis by Stein & Shakarchi and Visual Complex analysis by Tristan Needham.
Real analysis is an important course before Complex analysis as many of the concepts and proofs follow from it.
Also any suggestion for both real analysis and set theory? Thank you very much for the suggestion 🙏🏾
@@spiderjerusalem4009 For set theory I would recommend Jech's introduction to set theory.
For real analysis I recommend Rudin's Principles of Analysis if you want a thorough book. If you want an introductory book I recommend Stephen Abbott's Understanding Analysis.
May i know which software are you using to upload videos ?
How can we know where does a contour integration work?, like, does it need a special structure?, or anything like a series of conditions?, Im trying to lean it trough your videos and that would help a lot, thanks c:
Please someone who can tell me how to prove the formula sum (-1)*n/(t+n.pi)=csc(t) in 28:34
Link in the description
I have the following series would you be willing to take a look at it?
series((digamma(k+3/4)-digamma(k+1/4))/((4k+2)(4k+3))) sum from k= 0 to inf
This should have a closed form I believe, but I’m stumped on how one sums over digamma functions. You have one of the few channels with anything digamma related, would you be able to help?
Fascinating
I'll have a look at it in the morning. Just to be sure I don't forget it, send it to via the email given in the about section of the channel.
Email sent, I should also say that the email is related to a problem you’ve tackled on your channel before, I posted the link to that problem in the email.
7:52 not for t=0 what you assumed in the begining.
5:50 trying to pull a fast one on me? :D
Hi,
Can you please do the integral of (sin(x))^2 /(x^2*(x^2+1)) using contour integration Please?
Me and qncubed3 did a collab video on his channel where we solved that using contour integration and Feynman's trick. Check that out
@@maths_505 Thank you. Is it impossible to do this purely by contour integration without any prior application of trig substitution or Feynman's technique?
Yes qncubed3 solved it without Feynman's trick and used purely complex methods
@@maths_505 No he used trig substitution at the very beginning
I need your help❤
I'm syrian and im gonna graduate from high school soon and i have a good chance to study in the UK but I don't know what I should study
Im a person who love maths ,physics and machines I love to study a sector of engineering but I dont have a final decision I want u to advice me and consider me your small brother 😂
I'm sorry this comment has no thing to do with ur video but I really need help
note: I don't want to study maths in the university and thank u❤
Engineering....you can decide which discipline of engineering later when you're in university.
@@maths_505thank u so much ❤❤❤
Engel series whould also do the job
4 ways ruclips.net/video/qbr8GLNtdrA/видео.html
Are you aware of this 1909 article by GH Hardy in the Mathematical Gazette, a magazine for mathematics teachers? He gives a number of methods, giving marks to each.
math.harvard.edu/~ctm/home/text/others/hardy/sinx/sinx.pdf