a spectacular solution to the Basel problem (sum of 1/n^2 via a complex integral)
HTML-код
- Опубликовано: 22 авг 2018
- The infinite series of 1/n^2, i.e 1+1/2^2+1/3^2+..., actually converges to a special number, namely, pi^2/6. This is a very famous math problem known as the Basel Problem and it does have many different solutions. I want to thank my viewer, Zvi H., for providing a spectacular solution to find the sum of 1/n^2 via a complex integral.
💪 Support this channel, / blackpenredpen
🛍 Shop math t-shirt & hoodies: bit.ly/bprpmerch. (10% off with the code "WELCOME10")
#baselproblem #calculus #blackpenredpen
15:51, we learnt a lot about you 😂
I think he was talking about "i" haha. that play on words is hilarious. btw, thanks for sharing bprp!
THAT'S AN IMAGINARY OBSERVATION.
As soon as I heard it, I knew the top comment was about it
marbanak
I think it’s too *complex* of a message for us non-Rick-&-Morty fans to understand
Very sharp thinking!!
√-1 love this joke...
(i love this joke)
15:51 “I don’t like to be on the bottom, I like to be on the top.” 😂
Hahaha!
The best video on basel problem
Also, check out 3blue1brown's video on the same ;)
Existe uma prova muito legal no livro "tópicos de matemática elementar vol. 5" utilizando funções aritméticas.
I admire not the result but all the patience to write and explain every step :) if you want a problem with surprising result can you consider this one : take the polynomial (1+x+x^2)^n , note a_n the term of degree n , find an equivalent of it when n goes to infinity :)
Thank you!
i have done similar problem
Ok I admit, the end was really a surprise.
Video was uploaded 5 hours ago...you commented 2 days ago....
Are you from the future
I’m from the future of the future
I was from the past
Please post more content which link two entirely different maths together like this man!
That result is famous enough for there to be a proof wiki page on it with 7 proofs. This isn't one of them. You should add it to the list.
This isn't a proof, because the manipulations done with the series S aren't necessarily valid.
@@alxjones Thanks. You seem credible in what you said, in spite of the unfortunate man who shares your name. Sorry about that.
@@alxjones well if he proved the the series converges then everything is fine. The thing is that the series does indeed have a finite value so all the manipulations are valid
@@gregorykafanelis5093 the problem is that in the complex case you don't always have ln(xy)=lx(x)+ln(y)... the complex logarithm isn't to be taken carelessly, so maybe already in the third line, there is a mistake.
@@lukaskohldorfer1942 well he then have to say that the ln is restricted for only 2π to 0 angles. But then we get down the rabbit hole. Point being, this proof is a long way from being mathematically strict but it is a nice way to calculate the value of the integral
Also let's not forget the famous internet saying for mathematics
If the result is correct then the method must be correct. This time we can turn our heads to the other side as you have to admit this proof us truly beautiful not mathematically strict, but certainly has some beauty
that's an amazing integral! i will never stop learning from you
xamzx thank you!
blackpenredpen btw can you integrate cosx/x from pi/2 to +inf like you integrated sinx/x from 0 to +inf
Well done. Your joy is infectious.
And I had the joy of seeing yet another way on RUclips to calculate 1+1/2^2+1/3^2+1/4^2+...
And your way is quick and doesn't require much higher math.
It was Jacques Hadamard (who proved the prime number theorem, along with de la Vallee Poussin) who said that the shortest path to a truth in the real domain often passes through the complex plane.
With the end value of the Integral coming off so beautifully, this proof of the series expansion of pi^2/6 has been put into my math-related playlist where I keep all the beauty I find.
Keep it up!
#YAY
I'm afraid the proof looks wrong to me because the initial integral is not real (negative values inside the ln). But he needs it to be real for his argument.
Forget it, it's all fine, no negative values!
Very very nice! Brings to mind the old saying..(going back to the 1600's actually) "there is more than one way to skin a cat." ....Google the phrase....basically saying there is always more than one way to arrive at the same result.
Bernard Doherty I love this phrase!! Thank you. And I think that will be the perfect title of this video too! : )
Bernard Doherty I will change the "skin" to "brush" so that the cat lovers won't go after me. : )
Thanks for that
Mind blown!
Chris Hello : )
You can also use parseval's theorem with f(x) = x, which also gives the solution of the basel problem! Doing the same with x^2, x^3, et cetera gives all the positive even solutions for the zeta function (Zeta(2), Zeta(4), Zeta(6) et cetera)
Excellent video. Reminds me of complex analysis class I took in college. But that was so many years ago.
This is fantastic. One of my favourite videos/proofs yet
Did you just find the craziest way to prove that (pi^2)/6 identity?
I KNOW RIGHT????
Zvi did! : )
What less crazy way of proving the identity do you have in mind? The infinite product for sinc(x) and 3Blue1Brown's lighthouses are both pretty crazy to me.
JohnnyCrash
The simplest, should you say, ‘cuz the craziest thing we used was the complex (pun non-intended) definition of cosine in another problem, and everything was pretty straight-forward and self-explanatory....... compared to other proves
@@nicholasleclerc1583 the problem is that in the complex case you don't always have ln(xy)=lx(x)+ln(y)... the complex logarithm isn't to be taken carelessly, so maybe already in the third line, there is a mistake.
What a Brilliant way to prove this!!!❤️❤️
Proving one of the Best Equation in Maths in Best way!!!
So good! So many twists and interesting approaches. I was expecting just some random formula as a result.
Wow!! What a crazy way to get to the solution of the Basel problem!! Respect Blackpenredpen👏👏
Fascinating. Makes a lot of sense as you explain things. By the way I never thought of turning an integral into Macluaren or Taylor series if it is really difficult. Neat!
I really love mathematics. And I have been watching your videos and indeed, they have impacted my skills. I wish to meet you in life one day. Love your videos and hope to see more from you.
You and Dr. Peyam are killing it with these elementary proofs of pi identities!
That was awesome, all kinds of mathematical ideas connected and I love how Euler famous e^(i*pi) + 1= 0 was utilized. Finish the proof with basic algebra. Great fun!
Wonderful and quite elementary way to solve the Basilea problem.
Your videos rock! :)
rino strozzino thank you!
Very inspiring! People should love math by watching your video.
This is, in fact, one of the most beautiful videos i’ve ever seen
I love this way of solving this problem
Beautiful. Thank you for sharing!
What an amazing ending! Great video!
This video has made me so happy :)
Wow what a crazy cool Integral! Solving it seems fairly straightforward however where on earth did someone find out that this particular Integral leads to one of the most famous results in math? Either way, great video!
Thank you : )!!!
thats genius lol . this is really surprising and i really like the way you explain it too. good job
Schön gemacht - wie immer! Gratulation!!!
I really like this proof of the Basel problem. I have just one hiccup with the technicalities: when you evaluated the power series for log(1+z) and integrated it, don't you have to prove its absolutely convergent? I understand that it would've been too technical but a mention would have been nice. Either way, great video!
"don't you have to prove its absolutely convergent?"
That would be a shame since the series is not absolutely convergent.
@@martinepstein9826 Why? Is absolutely convergent different that just convergent?
@@createyourownfuture3840 They're different. The simplest example of a series that's convergent but not _absolutely_ convergent is
1 - 1/2 + 1/3 - 1/4 + ...
This converges to ln(2) but if you take the absolute value of each term you get
1 + 1/2 + 1/3 + 1/4 + ...
which diverges.
@@martinepstein9826 Oh...
@@martinepstein9826 Is that what he meant with “absolutely covergent”?
I really enjoy the way you're doing maths !
OMG this is so cool! I'm so happy that I found this video 😍😰
Great! Thank you for this 🙏
That was truely brilliant, wow
What a fantastic surprise to start watching an integration video and end up with a great proof!
Great proof! Very nice
Brilliant work
Loving your videos
Saw your videos ,BUT now you have a new subscriber
Love this channel
You were so nervous trying not making mistakes. It was hilarious you were so excited i really like it. Congrats
Obed Garza I was nervous trying to make sure I could fit everything on the board, as always : )
Tangentially related to this video, 3Blue1Brown has a fantastic video called "Why is pi here? And why is it squared? A geometric answer to the Basel problem" which shows a geometric proof that 1/1^2 + 1/2^2 + ... = π^2 /6 using lighthouses around circular lakes. Highly recommend checking this video out (along with the rest of that channel, his videos are awesome! :)
ruclips.net/video/d-o3eB9sfls/видео.html
Doug Rosengard I honestly feel that 3Blue1Brown’s logic in that video is kinda dodgy. Don’t get me wrong though, I love his videos. Especially his essence of Calculus Series.
Just curious where you disagreed with his logic in that video. It all seemed pretty well laid out to me. Also he links to a paper in his description "Summing inverse squares by euclidean geometry" which was the basis of the video
No doubt 3brown video on this topic is simply awesome as it triggers the intuition behind the very answer. However, this video is nothing less than amazing for all who loves maths.
In other words 3 brown Tries to answer in their every video, "why" certain things are the way it is.
This video tells how you get there.
I enjoyed both
This is pure magic 😍
that is actually insane. im blown away
Thank you so much. It's more than a math-game.
Me impresiona cada vez que veo estos videos. saludos.
Amazing! I tried plotting the graph of that just to see whether the real part will be zero. I couldn't comprehend it lol
The way he says super amazing..I am in just love with maths
Learn a lot from you keep teaching
Amazing proof, nice video sir.
I love everything about this video
Wow! That was perfect!!!
Dang you worked really hard on this video good job 👍
Gracias por tan interesante explicación. Saludos desde México.
You are the great in mathematics....congratulations....
That was very cool!
Great job! You could have shown that S for even integers (1/2^2 + 1/4^2 + 1/6^2+...) = pi^2/24, which follows from S = S (odd) + S (even), i.e., pi^2/6 = pi^2/8 + pi^2/24. It's beautiful
One loophole is where you show that \int( ln (e^(ix)) (from 0 to pi/2)= i*pi^2/8. Keep in mind that:
\int( ln (e^(ix)) = \int( ln (e^(i(x + 2*pi*n))
So, formally speaking, the result should be i*pi^2 * (1/8+ n) where n is an integer. Then you need to show that n must be 0.
This is really brillant and simple...
Amazing proof. I will show it to my teacher next year if he brings sum 1/n^2 up.
No doubt 3brown video on this topic is simply awesome as it triggers the intuition behind the very answer. However, this video is nothing less than amazing for all who loves maths.
In other words 3 brown Tries to answer in their every video, "why" certain things are the way it is.
This video tells how you get there.
I enjoyed both
Prettty, pretty amazing or awesome as you usually say. LIke we say in my pretty Nicaragua: you are a monster! You are a crack! Let me tell you that I follow you since the first time I found you in youtube. Have you thought in an analisis through any model for covid 19? I´ve seen a couple of them from Mathrock in Peru and Damian from Argentina. Man!!! you really love this beautiful science. Go on with that contagious enthusiasm!!!!!!!
wow that was awesome how he was able to fit all the steps on the board
19:08 LOL 🤣🤣🤣 the funniest part!
Absolutely Beautiful!
看到有的是通过给f(x)=x 傅立叶级数展开来给n平方分之一求和的。曹老师本次讲的这个过程也是很不错的!
Very good video !
I love the way you solved 9r very nice 🙂🙂🙂👍👍👍
I think you will get more subscribers fast. Wish you best luck.
This is just beautiful
The process of derivation is a piece of art.
Genius. Love it.
Wow...... nice one sir.... Thanks
Holy moley that's a beautiful result
Unknown Entity : )
GENIUS THIS IS AMAZING
BEAUTIFUL!
Beautiful!!
That was so awesome :D
Amazing result
@2:19 Isn't the complex log multi-valued? Would that change anything?
You restrict its definition so that it is single valued. The most common restriction is (-π, +π).
The path that the argument traces out doesn't cross a branch cut, so I guess we're OK
@@azmah1999 i prefer from [0,2π]
Very nice proof.
then you can write the original integral as the integral of ln(2) + the integral of ln(cosx), to get that the integral of ln(cosx) is -pi/2 * ln(2)!
So many results from this I am dying!😂
yes that's actually a pretty famous integral as well
What is integral ln(cosx). ?
Int 0 to pi/2 of ln(2) dx +Int 0 to pi/2 ln(sinx) =pi/2ln(2)-pi/2ln(2))
=0
this gives the answer as 0
Best vidéo on the net!
So much beauty in one formula
YES!!!!
Wow, just wow, this blew my mind...
That's so cool!
I wish you were available 10 years ago, you would have saved me from a lot of struggles
Beautiful!
thanks Dr
So amazing!!
Mmukul Khedekar thank you!!
Thanks!
Amazing one👍👍👍👍
Extraordinaire !
This video deserves 1M views and likes❤️
You solved basel problem with this amazing method? GREAT.
Awesome method