Apostol's Solution to the Basel Problem

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Жыл бұрын

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Пікірлер: 134

@benheideveld4617 Жыл бұрын

I have a pretty good idea what Michael Penn will say while taking his final breath…

@warrickdawes7900 Жыл бұрын

Should be on his gravestone!

@damascus21 Жыл бұрын

"Okay, nice. And that's a good place to stop."

@dethmaiden1991 Жыл бұрын

The first tombstone to have a tombstone written on it.

@dalitlegreenfuzzyman Жыл бұрын

🥺🥺🥺🥺😭

@marcushendriksen8415 Жыл бұрын

We need to make sure he has a chalkboard nearby so he can do one last filled square: ⬛️

@abrahammekonnen Жыл бұрын

5:27 Finicky thing: the determinant is already 2 not -2 because (1)(1) - (-1)(1) = 1 - (-1) = 2. But it doesn't matter because the absolute value.

@2kchallengewith4video 11 ай бұрын

I dont understand the part where he starts the iterated integral to 1/2

@tcmxiyw Жыл бұрын

One important fact my students tend to forget is sqrt(x^2)=|x|, so I got into the habit of saying sqrt(cos^2(x))=|cos(x)|, and then explaining why the absolute operation can be removed. Just removing the absolute value operation leads to errors when, say, the interval of integration contains regions in which a negative sign is needed to remove the absolute operation. Nice proof, though.

@leif1075 Жыл бұрын

A simpler proof is the infinite sum is equal to.the integral of 1/n squared dn so why nkt just do thst and the inegral.of that is 1/n..why not just do that instead of introducing x and y?

@briemann4124 Жыл бұрын

@@leif1075 because that isn’t correct. You can write a sum as an integral, but the differential would be d[n] and you have to handle the [n] appropriately. Your explanation skips over this entirely and wouldn’t give the proper value for the series.

@leif1075 Жыл бұрын

@@briemann4124 I forgot but you have mhlriply.by dn and take limit as dn goes to zero then it is correct..and an integral is basically just an infinite sum/prodict anyway though..

@briemann4124 Жыл бұрын

@@leif1075 it doesn’t change that your approach is incorrect. Yes, the series can be written as an integral using Riemann-Stieltjes. But you can’t evaluate it nearly as simply as you imply.

@leif1075 Жыл бұрын

@@briemann4124 it may not be simple but that doesnt meanmy approach is incorrect. You can write it as that integral.and then maybe worm.from tjere..why would say that is incorrect at all??

@gilamesh3417 Жыл бұрын

Jacobian Determinant is 2 even before absolute value!

@viktoryehorov4314 Жыл бұрын

yeah | a b | | c d | = ad - bc = 1 * 1 - (-1) * 1 = 2 (our case)

@NC-hu6xd Жыл бұрын

Wow REALLY ???? Peak midwitt comment

@viktoryehorov4314 Жыл бұрын

@@NC-hu6xd why are you so toxic? i appreciate prof. Michael Penn for his videos however, sometimes people might have no clue what’s going on, so my intention was only to emphasize the actual value of determinant, nothing more anyway I didn’t want to offense anyone, so never mind if i expressed it wrongly

@WhiterockFTP Жыл бұрын

this was a question on an analysis exam at my university once. of course the right integral transformation was stated as a hint - still i found it very cool to be proving something as famous as this on an exam. caught us a bit by suprise haha

@renesperb Жыл бұрын

This is a quite tricky way to solve the Basel-problem . Another way is to use Fourier series: if you expand the function f(x) =x *(Pi -x) in the interval (0 ,Pi) in cosine -functions you get f(x) = Pi^2/6 - (Cos2x/1 +Cos4x/4 +Cos6x/9 + ....). For x = 0 you have the solution of the Basel problem.

@backyard282 Жыл бұрын

wow this might be my favorite solution to basel problem so far, it doesn't involve any super crazy tricks

@ClaraDeLemon Жыл бұрын

The most sus aspect is perhaps changing the order of integration and summation, because you need to use lebesgue integrals and the like to prove it is a valid step, and that requires some work. But apart from that, all the steps are things that the average calc student could do by themselves, which is honestly amazing

@idjles Жыл бұрын

I discovered a solution to the Basel Problem involving NO calculus at all. It only uses Year 10 mathematics. It only uses sinx/x->1, (which can be proven with geometry), the zeroes of the sin and cos functions and Pythagorus Theorem. That's it. It gives π²/6.

@moutonso Жыл бұрын

@@idjles well let's see it !

@mustafaahmed9380 Жыл бұрын

@@idjles That sounds interesting. Where can I find such a solution?

@idjles Жыл бұрын

and if you calculate out the O(x⁴), O(x⁶) and higher terms, you can find all of the higher order Basel sums. I believe my proof is the simplest proof of Basel, and all higher orders, and can be followed by a child and you see exactly where the π², sum and 6 come from.

@jitzukinanaya4626 Жыл бұрын

another very elegant substitution can be utilized in the problem is let x=sin u/cos v, and y=sin v/cos u, and this integral becomes very simple.

@lightyagami1752 Жыл бұрын

How are you getting the determinant to equal negative 2? I just get 2. Even before the absolute value operation. (1)(1) - (-1)(1) = 2.

@Jack_Callcott_AU Жыл бұрын

@Light Yagami Me too! He made a slight miscalculation no doubt. 🧠

@silvio4386 Жыл бұрын

Yeah, it's 2.

@lightyagami1752 Жыл бұрын

@@silvio4386 Would be great if he issued a notice of correction. He skips a lot of steps, which is fine, I do that too - working mentally a lot, but you have to know you're right. If wrong, please remove confusion by issuing a correction.

@TimMaddux Жыл бұрын

I still have Tom Apostol’s Calculus I & II in my bookshelf. We called them “Tommy One” and “Tommy Two” when I was an undergrad.

@cutcrew2743 Жыл бұрын

I also have them from Dr Philip Curtis' UCLA class in 1960. I still remember that I was astounded that you had to prove the intermediate value theorem.

@roblangston319 Жыл бұрын

Same. Tom Apostol was one of my best instructors at Caltech.

@sniperwolf50 Жыл бұрын

I still have my used copy of Apostol's Calculus volume 1. It's a somewhat unorthodox text. It introduces integral calculus before differential calculus and then closes it off with linear algebra topics

@sergeipetrov5572 Жыл бұрын

Thanks a lot, Michael! I've learned one more way how to to determine that sum, owing to your brilliant explanation. That's just great!

@sachatostevin6435 Жыл бұрын

Another great video, Michael! Thanks :) I'm going to go read more about Jacobian thingys in multi-variable integral substitution stuff, because I'm not fully aware of them, but everytime I see you and Flammable-Maths do them, things seem to evaluate really nicely.

@stefanisraelssontampe996 Жыл бұрын

I going to use this result for small exersize in electrodynamics soon. Thanks for an inspiring post.

@alainleclerc233 Жыл бұрын

Another excellent solution to the Basel problem! Thanks Michael ! Another close way, is to use the substitution x=sin(u) and y=cos(v) instead the one proposed in the video as the integrand becomes simply 1 and the integral is just the Area of the integration domain, easy to compute as it is a triangle.

@goodplacetostop2973 Жыл бұрын

13:24

@birdbeakbeardneck3617 Жыл бұрын

just asking for a friend are you a mixhael alt account or a follower

@goodplacetostop2973 Жыл бұрын

@@birdbeakbeardneck3617 Not related at all. I’ve started this account for a joke, and I can’t stop doing it 😂

@vinvic1578 Жыл бұрын

@@goodplacetostop2973 and now you're an integral part of the community :D

@birdbeakbeardneck3617 Жыл бұрын

@@goodplacetostop2973 no asked cz u fast hh

@PeperazziTube Жыл бұрын

@@vinvic1578 Good Place To Stop make a big differential :)

@viktoryehorov4314 Жыл бұрын

also it's marvelous way to transform bound of integration (x,y) -> (u,v) 6:10, but i prefer other one in the video about average distance between two points P, Q in [0,1] * [0,1] square

@alanwhite3154 Жыл бұрын

Very cool 👍🏼

@Abhi-kr6df Жыл бұрын

Now i got the idea why euler is a genius. He solved in a most creative but easy way😂😅

@il_caos_deterministico Жыл бұрын

That’s a quite interesting problem close to the one that was presented in this video: Find the sum of 1/( (2n+1)(3n+1) ) Where n goes from 0 to infinity

@tubamazouz Жыл бұрын

Great !!

@qbqbqdbq Жыл бұрын

Apostol's analytic number theory book is brilliant.

@3x3-x3x-oXo Жыл бұрын

From 5:48 onwards I suggest directly setting u = sin(theta) v=cos(theta)sin(alpha) whose jacobian cancels out the integrand, and sends D to a kite-shape of area pi^2/12 in the theta-alpha plane. It's the same idea, expressed differently.

@Coldgpu Жыл бұрын

Never this method awesome 🔥🔥🔥🔥🔥🔥

@kalamardesk3666 Жыл бұрын

bravo

@Calcprof 15 күн бұрын

I had calculus from Apostol (with multivariable from Spivak's Calculus on Manifolds. 3 in the class got Math PhD's

@ecoidea100 Жыл бұрын

Beautiful video, the animations give an abstract but enlightening aura.

@koenth2359 Жыл бұрын

A great proof! Some minor details: The determinant is +2, not -2 but it does not matter. Also, the line of symmetry that is used is horizontal one, but this is not too confusing either, probably most viewers haven't even noticed.

@peterwaksman9179 Жыл бұрын

very nice

@mihaipuiu6231 Жыл бұрын

Beautiful demonstration. You are a very good speaker! You explain nicely, cleanly, clearly, w/ patience and logic.

@chrkts4191 Жыл бұрын

Uniform convergence gone boom

@Denosophem Жыл бұрын

Elementary solutions may be simple but there’s no thrill in using it

@replicaacliper Жыл бұрын

I would completely disagree

@rafabenjumea4683 Жыл бұрын

The double integral of 1/(1-xy) over the unit square can be expressed as the integral from 0 to 1 of (-1/x)*log(1-x). That integral must be π^2/6 but I have no idea how to prove it.

@TomJones-tx7pb Жыл бұрын

Much like this channel, Apostol's proofs in his books often made things way more computationally complicated and obtuse than they needed to be.

@Andbovi Жыл бұрын

Btw the determinant is two. But it doesn't change anything.🙂

@CM63_France Жыл бұрын

Hi, Amazing result, it means that there is no need to invoke the formula of Euler, so disputed: sin pi x /(pi x) = (1 - x^2) (1 - x^2/4) (1 - x^2/9) ...(1 - (x/n)^2)...

@peterclark5244 Жыл бұрын

I wouldn't say that the formula itself is disputed - it comes out quite naturally from Infinite product theory of complex analysis. It's more the usage without the proper rigorous justification (which didn't exactly exist at the time!)

@CM63_France Жыл бұрын

@@angelmendez-rivera351 you're right, I just wanted to say that Euler didn't worry much about convergence. But I do not question the genius of the find.

@CM63_France Жыл бұрын

@@peterclark5244 that's it.

@jellyfrancis Жыл бұрын

Can we have negative differential form 🤔

@mayonakao2488 Жыл бұрын

I’ll be sure to show this video to someone if they ever wonder what I mean after I tell them I like math.

@timothybrown8395 Жыл бұрын

I want to work out why this method fails for zeta(3), and I'm assuming it must or someone would have succeeded in it by now

@TheEternalVortex42 Жыл бұрын

I think there's no nice change of variables that causes the integrals to simplify.

@abrahammekonnen Жыл бұрын

3:00 Why do they need to be over the same interval? For example why can't y span from -1 to 1 instead?

@juyifan7933 Жыл бұрын

They dont, he misspoke a bit there. All he is doing is using Fubini Theorem, so it is enough that x^n.y^n be continuous in R^2, which it is. Maybe he was thinking about justifying the summation/integration change but mixed things up.

@peterclark5244 Жыл бұрын

@@juyifan7933 the interchange can be justified pretty easily by monotone convergence

@Jack_Callcott_AU Жыл бұрын

Such a nice proof. So many ways to solve the Basel problem. This seems one of the simplest. 🍉

@loopingdope Жыл бұрын

Why is he allowed to switch the integration and summation?

@scottmiller2591 Жыл бұрын

I see this is all true, but what motivated putting the original problem into a form with x and y in the first place (after the 2nd equals)? Yes, it looks like the integral of something, but why would anyone try that?

@robert-skibelo Жыл бұрын

Great video, as usual, but please don't pronounce Basel (the city) as if it was basil (the herb). The first syllable of Basel is pronounced "bah".

@bilalabbad7954 Жыл бұрын

Woww

@il_caos_deterministico Жыл бұрын

At min. 3:40 you really don’t explain why you can change order of integration and summation

@TheEternalVortex42 Жыл бұрын

By the dominated convergence theorem, since (xy)^n is bounded on [0,1]^2

@ladyloose Жыл бұрын

@@TheEternalVortex42 He should just state that. I was taught to justify that inversion pretty intensively back in the days.

@qdrtytre Жыл бұрын

He never does.

@il_caos_deterministico Жыл бұрын

Yes I know that, but just state it since it’s not a general thing one can do

@firemaniac100 Жыл бұрын

How do we now that sin(pi/6) = 1/2. Is there any simple proof?

@firemaniac100 Жыл бұрын

know

@firemaniac100 Жыл бұрын

here is a proof kzfaq.info/get/bejne/r5h6h9eUzLeZh6M.html

@firemaniac100 Жыл бұрын

One must take in consideration the definition of pi/2 as the first root of the cosine function

@timothybrown8395 Жыл бұрын

Are you familiar with the 30-60-90 right triangle and its properties?

@lightyagami1752 Жыл бұрын

Bisect an equilateral triangle with unit sides into two congruent right triangles and consider the trig ratios of each of the angles of one of the right triangles.

@funktorial 25 күн бұрын

i kinda dislike this solution because idk how one would ever come up with it

@RamsLiff Жыл бұрын

(u+v)(u-v) = u²-v², you got u²+v² somehow

@srikanthtupurani6316 Жыл бұрын

Most torturous topic for any beginning graduate student in pure math is general topology. words like box topology, lindeloff , limit point compact makes me feel so unhappy. We need enormous patience in case of general topology. It is all set theory and requires patience.

@otiswebb5783 Жыл бұрын

Nothing to do with the Basel problem but pls pls try this integral from 0 to inf (1/((x^2)-tanh(ln(x))) dx result is pretty nice

@charlienyikos1564 Жыл бұрын

Enbek aztán sok értelme volt phuuuuuu

@MGSchmahl Жыл бұрын

For anyone who is confused why 1/(n+1)² is rewritten as [x^(n+1)/(n+1)] etc: To be clear, any nonzero exponent of x (resp. y) could have been chosen, as long as they are independent of x (resp. y). x²⁰²² could have validly been chosen, but n+1 was chosen for convenience.

@Denosophem Жыл бұрын

i already liked commented and subscribed with no respones. sorry. something clearly doesnt work properly with algarithms

@andreamartelli823 Жыл бұрын

I would like to suggest a basic algebra problem. Show that 1/9801 = 0.00 01 02 03 … 96 97 99 00 01 … . (I didn’t forget the group 98). Then maybe find another example of this phenomenon.

@quantspazar6731 Жыл бұрын

Is the skipping of 98 because it's actually 98 99 100 but the 1 and 9 overlap so you get 98 100 00 but the 8 and 1 overlap so you get 99? Or is it another reason?

@Denosophem Жыл бұрын

The shakeup is evolution basically