Constructing interesting integrals

[@thomasmay6215]

Michael Penn preparing us for his proof of the Riemann Hypothesis.

[@R0M4ur0]

I guess he would start the video by saying "Here we're gonna look at a nice little theorem proof involving infinite sums. Our goal is to show that the real part of all the non trivial zeroes of this sum is equal to a half"

[@thomasmay6215]

@@R0M4ur0 Real part but yeah. And it will be a nice place to stop and accept the Millennium prize.

[@jerrysstories711]

Nope. He's preparing us for when he reveals his counter-example to the Riemann Hypothesis.

[@gustafa2170]

Funny thing is, he does end up solving it in a couple of years.

[@GKinWor]

@@gustafa2170 ???????

[@leif_p]

A couple people in the comments seem to think the Laplace transform is some special thing that they have to go "understand" before watching this video. You should have more confidence in yourselves. It's just an integral. If you've been watching this channel, you've seen much more complicated integrals. I think you all have the tools to understand the Laplace transform, and most of the other "transforms" already. Believe in yourself!

[@xCorvus7x]

Then, what makes it so special?

[@leastsignificantbit5069]

@@xCorvus7x its utility and pleasant to hear name, I guess

[@leif_p]

@@xCorvus7x Its properties, mainly linearity and the relatively simple transforms of the derivative, make it a useful tool to solve differential equations, esp. inhomogeneous ones that describe engineering systems where you're applying external forces/currents/etc.

[@tomatrix7525]

I’ve seen a lovely MIT lecture, years back, and on youtube, where the simple derivation is explained. It’s really nothing fancy but nice for ODEs

[@soranuareane]

Some constructive criticism for this particular video. Please turn off auto-focus (or lock focus) for the camera you're using. Once the blackboard is in focus, there shouldn't be any reason for the focus to adjust. The pan-zoom segments are jarring and disorienting. It makes us think there's something wrong or that we need to examine this result carefully, but you're using the device just to show the result. I suggest not having that in future videos (or, at least, ditching the transition and cutting to the result). Please keep up the videos! They're quite informative and fun to watch! This brings me back to the good ol' days of Calc 3 and DiffEq. Damn I enjoyed those classes...

[@JB-ym4up]

Bonus points for putting the logo over the final answer.

[@guilhermefranco2949]

13:20 what is going on his head at this very moment: "wait wtf? oh it's correct, nevermind"

[@random-td7tf]

Lol that’s a great observation

[@tiagobeaulieu1745]

Lollll I hadn't looked at his face, you're right.

[@mcwulf25]

I spotted that X & t switch. I think he meant to edit that out 😜

[@captainsnake8515]

I have a suggestion for a video topic you might enjoy: prove a closed form expression for: sum_r=0^\infty (binomialCoefficient(r+k, r)x^r by taking derivatives of the geometric series. It’s one of my favorite proofs in mathematics as it’s a very simple example of how we can use calculus and apply it to combinatorics, two fields which seem entirely different. I think it would fit your channel quite well, too.

[@tomatrix7525]

I like the idea of the zoom on a still image in areas, so we can see the full work while you write it so your back is not obstructing the view.

[@route66math77]

This is pretty rad. I like the idea of developing tools to construct your own integrals!

[@goodplacetostop2973]

18:07 Is that a Goron from Zelda Breath of The Wild (in the thumbnail)?

[@MichaelPennMath]

Yeah, he helped me “construct” the integral!

[@basambinsohailraja1801]

Oh wow found you

[@random-td7tf]

What ? You r 3 weeks early again ?

[@goodplacetostop2973]

@@random-td7tf Told you, I’m the time wizard 😛

[@adambascal]

@@random-td7tf this video was unlisted for a while (I actually watched it yesterday)

[@dozi3r]

just found this guy, reinvigorated my love for math

[@mcwulf25]

Blackpenredpen did that for me. And he recommended Michael's channel.

[@SimpMaker]

Thanks

[@nebulasy8]

At the very end, the channel logo obscured the final result.

[@berzerksharma]

I have saved this to watch later, Will comeback after learning Laplace transformations

[@UltraMaXAtAXX]

Integrarion by parts. Done.

[@shanmugasundaram9688]

There is no generala formula for finding zeta values for integers although there is one for even Integers involving Bernoulli's numbers.This problem gives zeta values in integrals of trigonometric functions although not in closed form.For m=3 the answer is zeta(3)/4 for the given problem.

[@TheDJSyaheer]

This is the coolest I've seen in a while...can you provide the name of this integral identity? I've tried searching for it

[@nombreusering7979]

Time traveling be like

[@geilomatico7607]

the zooming is weird...

[@kqp1998gyy]

Is not effective and redundant especially with the focus issues after it.

[@levonnigogoosian7547]

I agree

[@mcwulf25]

As is the focusing.

[@AnCoSt1]

The zooming looks weird but there were focus issues and my guess is the zooming happens in parts where the focusing was especially bad.

[@mcwulf25]

@@AnCoSt1 Yes I noticed there was a chronological connection between the two. Interesting that the zooming went to the top corner, and what he was describing. My guess is that it was deliberate but with unintended consequences.

[@AnCoSt1]

At 11:00, the representation shares a similar pattern to the Gamma function. Am I right to see something there, or does that not pan out to anything with the unpleasant denominator?

[@JXS63J]

Hold it hold it ONE second!!! Where did you get that hoodie?

[@meiwinspoi5080]

the camera lens is searching focus once too often. keep your camera at manual focus with focus at the board. adjust the aperture to f4 to get a bit of depth of focus. adjust the iso to push the EV to +1 or thereabouts and at the same time make sure the exposure doesnt cross 80%. please avoid the zooming into a portion of the board. i use a white board for my videos and use this method to ajdust the loghting.

[@kevinmartin7760]

The EV adjustment for a whiteboard would probably not be applicable to a blackboard. Though I agree that the zooming into part of the board isn't useful. Things sink in just a bit better if I see him writing them as he is speaking.

[@Tezhut]

@@kevinmartin7760 I think it's likely that he added the zooms in later because his camera went out of focus and we wouldn't be able to see what was happening. Don't you think?

[@CM63_France]

I aggree, not necessary to zoom. However, I thing the problem of focus adjusting is another thing.

[@meiwinspoi5080]

zooming in is unnecessary. keep it as real classroom as possible. agree ev adjustment for blackboard wont work. the warm tone is making the overall scene a bit sombre. dont young minds prefer a brightly lit classrooms? isnt it better to keep the temp neutral at 5000k. left to myself i will present a bright environs and mimic a naturally bright classroom atmosphere.

[@meiwinspoi5080]

@@angelmendez-rivera351 well, i m from india a tropical sunny country. micheal penn’s videos are essentially targeting senior secondary maths students (atleast highly motivated math nerdy senior secondary students that is). senior secondary students are in the age group of 16-18 in general. 99.999% of classrooms in india are naturally lit airy open classrooms and it is natural for students to expect a natural lighting in the learning videos. my opinion is specific to my country. india could be a major subscriber base for any math channel with its 500 million+ youngsters learning in english as a national common language (faar higher than china in terms if english reach and definitely larger than US and Europe put together). so if any one targeting india it is better to be bright and colourful.

[@CM63_France]

Hi, 12:12 , and two more instants before : focus adjusting problem, 13:10 : dt = dx / (1-x) and not 1 / (1-x) For fun: 2 "so let's may be go ahead and", 1 "let's may be go ahead and".

[@JalebJay]

I'm not sure what you were trying to do at 2:27, but felt somewhat wierd with abrupt cut/zoom in

[@ashimdey1004]

The angles of the triangle ABC satisfy the condition 2∠A+∠B=∠C. Inside this triangle, the point K is chosen on the bisector of the angle A such that BK=BC. Prove that ∠KBC=2∠KBA.

[@ashimdey1004]

@Let's play minecraft Yes

[@mikeschieffer2644]

How do we know -1 < e^(-t) < 1 so that we can use the geometric series representation?

[@boristerbeek319]

We're looking at the domain 0 < t < infinity. Since e^(-t) is strictly (monotonically) decreasing in t, its maximum occurs at the lowest possible value of t, which is t = 0. Hence the maximum value is e^(-0) = 1, so we have e^(-t) < 1 for all 0 < t < infinity. Also, 0 < e^(-t) for all t, so that in particular -1 < e^(-t) < 1.

[@calamariattack]

t is a dummy variable inside an integral, and the domain of the integral is from t -> 0 and t -> +infinity, which means that e^(-t) goes from 1 to 0 on this domain, so the geometric series limit is valid

[@dicksonchang6647]

because t is in the interval (0, infinity)

[@mathmatikapret5712]

Is this function continues in this range(0 to pi/2)? Then we can evaluate intergration sir?

[@divergentmaths]

Hi Michael, can you solve this integral? int_{0}^{+∞} sin(t ln2)/(e^{2t pi}-1) dt

[@user-A168]

good

[@mcwulf25]

Ok I will be picky. Missed out the dx at 12.08 but managed to gloss over that when using the result .

[@pacolibre5411]

Proof that there is no such thing as “the wrong u-substitution”

[@The1RandomFool]

I worked out the integral in the thumbnail to 1/4*zeta(3) before watching the video.

[@montahd7530]

at 9:29 you applied the sum of the geometric series and that is wrong! U should be at the interval of [-1..1] but the domain of the integral is [0..inf[ even the exponential function have the domain of [0..inf[ ! the geometric sum should be only defined at [-1..1] other wise the sum will diverge! and can't be written to (1/1-U)

[@ricardocavalcanti3343]

u = e^(-t), so u is in the interval (0,1) when t is in the interval (0, infinity).

[@montahd7530]

@@ricardocavalcanti3343 no U=e^(-t) have a domain of ]0.. +infinity[ not including the (-1..0)! you summing terms that doesn't exist in (-1..0)! the value of e^(-t) can't even be negative! that is what i'm talking about!

[@MoodyG]

Hello Michael, I am stuck trying to evaluate the integral of e^(-x-a/x)/(b+c/x) from 0 to +inf, where a, b, and c are constants. Help a bro out here :D

[@whatisthis2809]

theres comments from 3 weeks ago but video uploaded 3 minutes ago well i guess im happy i spent 4 minutes googling memes

[@nebulasy8]

Hi Michael, You make the best and most interesting math videos on KZbin. Having said this: 1. Fix the focus of your camera, so that it stops guessing the focal length as it shoots, since neither the camera nor the object (you and the board) are moving. The scene is not dynamic, but fixed. 2. The lighting is too orangey. I doubt the scene is actually that orangey and warm in real life. If it is, the lights color isn’t right, if it is not, perhaps don’t fix it in post production. 3. The cut scenes are distracting and confusing. Why? Naturally when in lecture, our field of vision always includes the lecturer who’s writing this, but in the cut scenes, the lecturer is missing. The makes the brain wonder where’s the missing element and makes one confused. Please don’t use them. Thanks for the great teaching videos!

[@mcwulf25]

My eyes went out of focus a couple of times 🤔. A couple of minor mistakes there but I won't be picky. Interesting what happens when you play around with the variables. Who would have thought to chuck in a Laplace Transform into that formulation?

[@matthias7790]

Did you actually play breath of the wild? :)

[@MichaelPennMath]

I did. I completed all 120 shrines plus the extra DLC content. I was a bit obsessed with this game for a while.

[@matthias7790]

@@MichaelPennMath I still chase the 900 Koroks. It's a mess I can tell.

[@boristerbeek319]

@@matthias7790 poor soul! Have my condolences in advance ;)

[@goodplacetostop2973]

@@matthias7790 You mad man 😂 I gave up on these. Great game nonetheless

[@adambascal]

@@goodplacetostop2973 one of my favourite games :)

[@romajimamulo]

17:57 can you move the sub to my channel circle so it's not over the equation?

[@stephenbeck7222]

In this case, it’s the same as the equation before but with the change to the outside that you can see. In general, he should add an “outro” roll of some sort so he can still have the pop up cards without blocking the board.

[@natepolidoro4565]

That thumbnail tho

[@ikarienator]

Missing a dx at 12:09.

[@PenguinPat]

I wonder how many of my math professors are Zelda fans.

[@xCorvus7x]

The first Zelda game was published 1986, so maybe more than you think. Besides, that number is probably steadily increasing.

[@harshanand127]

❤️

[@jerrysstories711]

The zooming in is annoying and distracting. Please just let us see the board, and we'll direct our eyes where we need to. You already have a great way of presenting the material, just like in a classroom, except I can see the whole board clearly and don't have to wait for boards to be erased. Please stop experimenting with embellishments.

[@joeg579]

it's good to let him know that the zooming in is unnecessary, but we should be encouraging (educated) trial-and-error experimentation with content creators, not telling them to cease and forever use the same format

[@-UMT-VIGNESHKRISHNAS]

13:09 minus symbol missing sir for dt

[@xCorvus7x]

Why? He is deriving it from the equation t = -ln(1-x) .

[@stephenbeck7222]

Check your chain rule.

[@random-td7tf]

Michael how do you get so much free time ? (I mean you’re a prof but still post videos almost daily )

[@stephenbeck7222]

Many of the videos are directly related to class content. I think he likes to ‘flip’ his classes so he not only posts videos for students to watch but also has class discussion time. So, I have no idea where he gets the time including his rock climbing hobby and family.

[@keshavb3128]

the video thumbnail was creepy. speaking of which, I think Professor Penn is now a millionaire.

[@wjrasmussen666]

Oh man, I was just talking with a bunch of people about this while waiting for a bus.

[@anonimmors1925]

Nice, after some caluclations it was looking like Bernoulli series but nevermind, intresting solution.

[@cariboubearmalachy1174]

What's with the weird camera work today?

[@kevinmartin7760]

I would simplify 2*(-2^(m-1)) to -(2^m) Edit: After mcwulf25's comment I realized the simplified form should be -((-2)^m). My original simplification would not alternate in sign.

[@mcwulf25]

Not the same. If m is odd then 2* (-2)^(m-1) is positive but -(2^m) is negative. But we could have put 2^m * (-1)^(m-1), which I prefer.

[@kevinmartin7760]

@@mcwulf25 Woops, of course the base of the power must remain negative so the sign alternates. I meant -(-2^m)

[@gaulindidier5995]

Gorons are cute.

[@birdboat5647]

more botw content give the people what they want

[@BoringExtrovert]

I don't like the zooming tbh

[@GKinWor]

botw fan?

[@klementhajrullaj1222]

The Devil! 😀😉 ...

[@KishanSingh-fv9qj]

I don't know anything about Laplace transforms so couldn't understand the video and very sad for tha :(

[@stephenbeck7222]

The Laplace transform is a minor part of the video and he gave you everything you need to know about it with the “recall” and the “Tool”. The “recall” is just a definition of an operation that takes a function and turns it into a weird integral. Actually the main idea of the video is doing transformations on the Reimann Zeta function, a very important and famous function, but also one which you have no need to understand at all for the purposes of following the calculus here.

[@mcwulf25]

You can skip over the proof of the transform - it's probably in a list of transforms anyway. He is only using the formula as a stepping stone, like he used the trig substitution. You don't need to understand trig to apply the substitutions.

[@ashimdey1004]

Please solve the problem which I have sent you!