a spectacular solution to the Basel problem (sum of 1/n^2 via a complex integral)

Рет қаралды 310,520

5 жыл бұрын

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.
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Пікірлер: 585

@abidurrahman4641 5 жыл бұрын

15:51, we learnt a lot about you 😂

@jameswilson8270 5 жыл бұрын

I think he was talking about "i" haha. that play on words is hilarious. btw, thanks for sharing bprp!

@marbanak 5 жыл бұрын

THAT'S AN IMAGINARY OBSERVATION.

@WhattheHectogon 5 жыл бұрын

As soon as I heard it, I knew the top comment was about it

@nicholasleclerc1583 5 жыл бұрын

marbanak I think it’s too *complex* of a message for us non-Rick-&-Morty fans to understand

@tamirerez2547 5 жыл бұрын

Very sharp thinking!! √-1 love this joke... (i love this joke)

@v_saaam 5 жыл бұрын

Ok I admit, the end was really a surprise.

@jibran8410 5 жыл бұрын

Video was uploaded 5 hours ago...you commented 2 days ago.... Are you from the future

@applealvin9167 3 жыл бұрын

I’m from the future of the future

@v_saaam 3 жыл бұрын

I was from the past

@Alex-xc9sf 5 жыл бұрын

15:51 “I don’t like to be on the bottom, I like to be on the top.” 😂

@johnrodonis4186 3 жыл бұрын

Hahaha!

@Timorftw 5 жыл бұрын

The best video on basel problem

@cocoa1996 5 жыл бұрын

Also, check out 3blue1brown's video on the same ;)

@silverbladeii 4 жыл бұрын

Existe uma prova muito legal no livro "tópicos de matemática elementar vol. 5" utilizando funções aritméticas.

@firashadjtaieb6730 5 жыл бұрын

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 :)

@blackpenredpen 5 жыл бұрын

Thank you!

@verainsardana 5 жыл бұрын

i have done similar problem

@Sam_on_YouTube 5 жыл бұрын

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.

@alxjones 5 жыл бұрын

This isn't a proof, because the manipulations done with the series S aren't necessarily valid.

@Sam_on_YouTube 5 жыл бұрын

@@alxjones Thanks. You seem credible in what you said, in spite of the unfortunate man who shares your name. Sorry about that.

@gregorykafanelis5093 5 жыл бұрын

@@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

@lukaskohldorfer1942 5 жыл бұрын

@@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.

@gregorykafanelis5093 5 жыл бұрын

@@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

@DarkMage2k 5 жыл бұрын

Please post more content which link two entirely different maths together like this man!

@Achill101 3 жыл бұрын

Well done. Your joy is infectious. And I had the joy of seeing yet another way on KZfaq to calculate 1+1/2^2+1/3^2+1/4^2+... And your way is quick and doesn't require much higher math.

@johnnath4137 3 жыл бұрын

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.

@xamzx9281 5 жыл бұрын

that's an amazing integral! i will never stop learning from you

@blackpenredpen 5 жыл бұрын

xamzx thank you!

@xamzx9281 5 жыл бұрын

blackpenredpen btw can you integrate cosx/x from pi/2 to +inf like you integrated sinx/x from 0 to +inf

@johnnycrash1624 5 жыл бұрын

Did you just find the craziest way to prove that (pi^2)/6 identity?

@rot6015 5 жыл бұрын

I KNOW RIGHT????

@blackpenredpen 5 жыл бұрын

Zvi did! : )

@martinepstein9826 5 жыл бұрын

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.

@nicholasleclerc1583 5 жыл бұрын

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

@lukaskohldorfer1942 5 жыл бұрын

@@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.

@gergodenes6360 5 жыл бұрын

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

@dustinbachstein3729 3 жыл бұрын

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.

@dustinbachstein3729 3 жыл бұрын

Forget it, it's all fine, no negative values!

@edmundwoolliams1240 5 жыл бұрын

This is fantastic. One of my favourite videos/proofs yet

@martinepstein9826 5 жыл бұрын

You and Dr. Peyam are killing it with these elementary proofs of pi identities!

@pimcoenders-with-a-c1725 5 жыл бұрын

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)

@arequina 5 жыл бұрын

Excellent video. Reminds me of complex analysis class I took in college. But that was so many years ago.

@ripansharma5259 5 жыл бұрын

Wow!! What a crazy way to get to the solution of the Basel problem!! Respect Blackpenredpen👏👏

@zwz.zdenek 5 жыл бұрын

So good! So many twists and interesting approaches. I was expecting just some random formula as a result.

@willyh.r.1216 4 жыл бұрын

Very inspiring! People should love math by watching your video.

@cliffhanger4930 5 жыл бұрын

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!

@MrNicolas609 5 жыл бұрын

This is, in fact, one of the most beautiful videos i’ve ever seen

@krrishmaheshwari4860 3 жыл бұрын

What a Brilliant way to prove this!!!❤️❤️ Proving one of the Best Equation in Maths in Best way!!!

@ishakbasiru8561 3 жыл бұрын

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.

@rinostrozzino6467 5 жыл бұрын

Wonderful and quite elementary way to solve the Basilea problem. Your videos rock! :)

@blackpenredpen 5 жыл бұрын

rino strozzino thank you!

@pspmaster2071 5 жыл бұрын

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!

@ManiFunctor 5 жыл бұрын

Mind blown!

@blackpenredpen 5 жыл бұрын

Chris Hello : )

@bernarddoherty4014 5 жыл бұрын

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.

@blackpenredpen 5 жыл бұрын

Bernard Doherty I love this phrase!! Thank you. And I think that will be the perfect title of this video too! : )

@blackpenredpen 5 жыл бұрын

Bernard Doherty I will change the "skin" to "brush" so that the cat lovers won't go after me. : )

@scathiebaby 5 жыл бұрын

Thanks for that

@hansulrichkeller303 Жыл бұрын

Schön gemacht - wie immer! Gratulation!!!

@soupe2000 5 жыл бұрын

I love this way of solving this problem

@linusschwan6299 5 жыл бұрын

What an amazing ending! Great video!

@radiotv624 5 жыл бұрын

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!

@blackpenredpen 5 жыл бұрын

Thank you : )!!!

@celkat 4 жыл бұрын

Beautiful. Thank you for sharing!

@grindpalm Жыл бұрын

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

@benoist13 4 жыл бұрын

I really enjoy the way you're doing maths !

@titan4267 5 жыл бұрын

thats genius lol . this is really surprising and i really like the way you explain it too. good job

@joshuacoppersmith 5 жыл бұрын

What a fantastic surprise to start watching an integration video and end up with a great proof!

@matanfih 5 жыл бұрын

Saw your videos ,BUT now you have a new subscriber

@obedgarza6236 5 жыл бұрын

You were so nervous trying not making mistakes. It was hilarious you were so excited i really like it. Congrats

@blackpenredpen 5 жыл бұрын

Obed Garza I was nervous trying to make sure I could fit everything on the board, as always : )

@fiNitEarth 5 жыл бұрын

OMG this is so cool! I'm so happy that I found this video 😍😰

@DarkMage2k 5 жыл бұрын

Amazing! I tried plotting the graph of that just to see whether the real part will be zero. I couldn't comprehend it lol

@nickm1902 4 жыл бұрын

This video has made me so happy :)

@faith3174 5 жыл бұрын

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!

@martinepstein9826 3 жыл бұрын

"don't you have to prove its absolutely convergent?" That would be a shame since the series is not absolutely convergent.

@createyourownfuture3840 2 жыл бұрын

@@martinepstein9826 Why? Is absolutely convergent different that just convergent?

@martinepstein9826 2 жыл бұрын

@@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.

@createyourownfuture3840 2 жыл бұрын

@@martinepstein9826 Oh...

@MaximQuantum 2 жыл бұрын

@@martinepstein9826 Is that what he meant with “absolutely covergent”?

@mortezamodarres2470 5 жыл бұрын

That was truely brilliant, wow

@shardumachal 5 жыл бұрын

The way he says super amazing..I am in just love with maths

@shashikumar7890 5 жыл бұрын

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

@yds6268 3 жыл бұрын

Great proof! Very nice

@stefanoctaviansterea1266 5 жыл бұрын

Amazing proof. I will show it to my teacher next year if he brings sum 1/n^2 up.

@sumitprajapati821 4 жыл бұрын

19:08 LOL 🤣🤣🤣 the funniest part!

@aashsyed1277 3 жыл бұрын

I love the way you solved 9r very nice 🙂🙂🙂👍👍👍 I think you will get more subscribers fast. Wish you best luck.

@superrobust 2 жыл бұрын

Thank you so much. It's more than a math-game.

@semi8883 5 жыл бұрын

that is actually insane. im blown away

@unknown....789 3 жыл бұрын

Great! Thank you for this 🙏

@adarshagrawal3555 4 жыл бұрын

Amazing proof, nice video sir.

@i-smartchamps3965 4 жыл бұрын

Loving your videos

@coolzo2524 2 жыл бұрын

Dang you worked really hard on this video good job 👍

@stanentertainment7711 3 жыл бұрын

Love this channel

@sunildey5887 5 жыл бұрын

Learn a lot from you keep teaching

@garethxue8938 Жыл бұрын

Brilliant work

@Matthew-tu2jq 5 жыл бұрын

This is pure magic 😍

@spartacus8875 3 жыл бұрын

You are the great in mathematics....congratulations....

@benattitude 2 жыл бұрын

I love everything about this video

@AngadSingh-bv7vn 3 жыл бұрын

wow that was awesome how he was able to fit all the steps on the board

@ShenghuiYang 5 жыл бұрын

The process of derivation is a piece of art.

@__-1234 3 жыл бұрын

This is really brillant and simple...

@Bodyknock 5 жыл бұрын

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! :) kzfaq.info/get/bejne/mpOfZshzntnKnKc.html

@BrotherSquid 5 жыл бұрын

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.

@Bodyknock 5 жыл бұрын

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

@shashikumar7890 5 жыл бұрын

@Salvador964 2 жыл бұрын

Gracias por tan interesante explicación. Saludos desde México.

@alejandronunezmontero2469 5 жыл бұрын

No sé inglés .Pero te entiendo .

@patmark5886 5 жыл бұрын

Best vidéo on the net!

@comingshoon2717 4 жыл бұрын

Me impresiona cada vez que veo estos videos. saludos.

@___xyz___ 5 жыл бұрын

Holy moley that's a beautiful result

@blackpenredpen 5 жыл бұрын

Unknown Entity : )

@leviszhou711 3 жыл бұрын

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！

@oscarfu9556 4 жыл бұрын

This video deserves 1M views and likes❤️

@mathmaths8380 5 жыл бұрын

So much beauty in one formula

@blackpenredpen 5 жыл бұрын

YES!!!!

@efeguleroglu 5 жыл бұрын

Wow! That was perfect!!!

@mokouf3 4 жыл бұрын

You solved basel problem with this amazing method? GREAT.

@matthewstevens340 5 жыл бұрын

Personally I love the Taylor series approach using Sin(x)/x and treating it as a polynomial to solve the Basel problem, but this is amazing!

@antoinemorrier9858 2 жыл бұрын

That's the way Euler had an intuition about the result. But it is not rigourous enough ^^

@Liesse_SportSante 4 жыл бұрын

Very good video !

@user-pw6qe7ur4q 2 жыл бұрын

BEAUTIFUL!

@inesantoniosanchezgutierre664 4 жыл бұрын

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!!!!!!!

@santiagoerroalvarez7955 5 жыл бұрын

This is just beautiful

@andreapaps 3 жыл бұрын

That was so awesome :D

@firecorex994 3 жыл бұрын

Beautiful!!

@wenhanzhou5826 5 жыл бұрын

Wow, just wow, this blew my mind...

@anasettajani8123 5 жыл бұрын

I wish you were available 10 years ago, you would have saved me from a lot of struggles

@Aldron6 5 жыл бұрын

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)!

@nischay4760 5 жыл бұрын

So many results from this I am dying!😂

@faith3174 5 жыл бұрын

yes that's actually a pretty famous integral as well

@manishkumarsingh3082 5 жыл бұрын

What is integral ln(cosx). ?

@SanjeevKumar-js4mu 5 жыл бұрын

Int 0 to pi/2 of ln(2) dx +Int 0 to pi/2 ln(sinx) =pi/2ln(2)-pi/2ln(2)) =0

@gaunterodimm7219 5 жыл бұрын

this gives the answer as 0

@pedjolinko Жыл бұрын

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.

@vinayakrao6687 4 жыл бұрын

Wow...... nice one sir.... Thanks

@mmukulkhedekar4752 5 жыл бұрын

So amazing!!

@blackpenredpen 5 жыл бұрын

Mmukul Khedekar thank you!!

@rickybobby5584 5 жыл бұрын

Very nice proof.

@Swapnil5 5 жыл бұрын

Absolutely Beautiful!

@chronicsnail6675 4 жыл бұрын

GENIUS THIS IS AMAZING

@elliottmanley5182 5 жыл бұрын

Genius. Love it.

@nathanisbored 5 жыл бұрын

never knew you could prove it that way, wow

@BrunoVisnadi1 5 жыл бұрын

I was about to link you this video in Discord when I saw your comment. This proof was so beautiful.

@unflexian 5 жыл бұрын

Meeting you again, it seems? I guess it is a small world after all. Love your content!

@gregoriousmaths266 4 жыл бұрын

Which discord

@aweebthatlovesmath4220 2 жыл бұрын

Amazing result

@i-smartchamps3965 4 жыл бұрын

Love this one I thought we had to solve with zeeta functinons

@salifdiallo4627 4 жыл бұрын

Extraordinaire !

@saksham1919 4 жыл бұрын

how could one dislike a video so good

@moskthinks9801 5 жыл бұрын

Zvi H. (or whomever told him) is the next Euler! Out of the random why is the time length sqrt(5)=2.236...?

@blackpenredpen 5 жыл бұрын

M. Shebl : )

@MrBoubource 5 жыл бұрын

Yay! I figured the result at 17:17 with my head and a piece of paper! But I didn't you the fact that it had to be real, and computed the sum of the inversed odd squares with series :( That is a decent way of showing that the sum of the inverse of squares from 0 to infinity is pi^2/6 ahah (I assumed that when I used series.)