Researchers thought this was a bug (Borwein integrals)

[@3blue1brown]

The next video on convolutions and their relationship to FFTs is out! kzbin.info/www/bejne/gaa7m6p4abGtiaM

[@Math4e]

What a time to be alive! Thank you. Also what a good timing! The last episode of Veritasium was also about the fast Fourier transform and there, Derek mentioned you! :-)

[@Tfame]

Bless your soul! Your videos are the only thing that bring me sanity

[@stevenspencer306]

I like the hint about multiplying large numbers being related to convolution. It took me until well after grad school to realize that the long multiplication I was taught in second grade, was actually a convolution.

[@Math4e]

@@stevenspencer306 Really? Seems interesting!

[@Pilchard123]

@@Math4e Are you holding on to your papers?

[@smartereveryday]

This is amazing. I even love the way you visually explained moving averages.

[@yongdetao7005]

hi

[@GamerDS76]

E

[@gabor6259]

Hi, Destin.

[@rgw5991]

i hate averages

[@anoobis117]

Laminar transform

[@marshallmykietyshyn4973]

One of the main problems I have in making presentations is that I always try to make them like a story, avoiding spoilers so that everything leads up to the interesting take-home point, but you don't know what is coming until I get to it. This channel demonstrates why that's a flawed way of thinking for educational purposes. It's so much easier to follow along with these explanations knowing where they are going. The explanation at 4:22, while seeming like spoilers to me in the moment, was actually extremely helpful.

[@CmdEngineer]

yes

[@goofyrice]

If you want to guide someone to a destination, show them the whole map before giving individual instructions. That way if they make a wrong turn, they can have some sense that they’re going the wrong direction. Landmarks and reviewing the map partway through are important for humans learning how to get somewhere.

[@samarendra109]

I don't think that's a spoiler, rather that's a hook. Like movies doing "you must be wondering how I got here" type. Hooks are really important in story telling as that builds the interest in the subject matter. The actual Spoiler in this case is the relationship between the two graphs via Fourier Transform.

[@Rodhern]

Don't worry. It is just two ways of making presentations. While Grant does claim his is superior (in some of his other videos), not everyone agrees. I suspect I would enjoy your storytelling style.

[@meiliyinhua7486]

even in storytelling: foreshadowing or even straight up giving answers ahead of time to give a sense of dramatic irony is a useful tool for creating hitchcock-esque suspense in a situation where surprise is not sufficient for making the story good. It's one thing to know *what* happens, another to see *how* it happens, and sometimes knowing what happens makes you wonder how

[@brianparisien9262]

I'm a retired electro-geek who last studied this stuff over 40 years ago. Having just discovered this channel, I wish I'd had this resource prior to slogging through the computational mechanisms available to us at that time. These verbal and graphical explanations are absolutely fabulous, and I foresee hours of enjoyable education in my future with a cup of coffee in one hand, these videos on my side screen, and a spreadsheet in front of me. Thank-you!

[@jaewok5G]

I remember turning my homework paper to 'landscape' to solve Fourier transforms 'by hand' in order to fit them on one line.

[@anthonybasile6079]

That's the most EE thing I've heard in a while, and I work as a plant electrician....

[@davewaterworth8846]

I thought exactly the same thing, I studied undergraduate electrical engineering 30 years ago which was very heavy on laplace and fourier transforms, and convolutions. This video would have helped me understand them infinity better back then!

[@Phosfit]

Have you lived a happy life?

[@Pxrchi]

​@davewaterworth8846 always been a fan of 3b1b, even back in middleschool I remember watching his videos on topics that I could barely understand, but yet still amazed me. Currently working on some projects regarding DSP, specifically convolution and spectral analysis, so I just had to click on this video. But man, even with a good understanding of these mathematical concepts, I've been stuck on applying it in DSP discretely, especially convolution. Never realized the amount of things that the fourier transform is used for.

[@istvankertesz3134]

fun fact: Bor means wine in Hungarian, and Wein means wine in German, so if you translate it, it's the winewine integral.

[@AlexanderAue]

I was thinking the same thing 😅 Szeretnél velem inni egy pohár bort?

[@evandickinson3254]

Pretty sure Wein means Vienna on German, no?

[@istvankertesz3134]

@@evandickinson3254 That's Wien, not wein.

[@bable6314]

@@istvankertesz3134 And people say English is a strange language.

[@nikkiofthevalley]

​@@bable6314It's wee-en vs w-ine (I think, I haven't spoken German in years)

[@M4DA.]

For the very first time, the bug actually WAS a feature

[@aze4308]

imagine copying lol

[@MichaelPohoreski]

Games have been doing this for _decades._

[@aesign]

Lol the universe is like mojang

[@vigilantcosmicpenguin8721]

The developers of Real Life™ just left a bug in production and hoped no one would notice.

[@pwnmeisterage]

Math and physics seem to contain some undocumented easter eggs.

[@diarya5573]

"a tiny positive number my computer couldn't compute in a reasonable amount of time" You should do it the Matt Parker way, put it up on the internet and people will improve your code by a factor of millions in a matter of days!

[@3blue1brown]

Good point! If anyone out there has a deeper knowledge of computer algebra systems, I'm all ears.

[@PMA_ReginaldBoscoG]

@@3blue1brown Ask Matt Parker himself to do so 😂

[@Восьмияче́йник]

The answer is π times 692235940415362523136988414491285998468620532382124599554066975879968202372479018941687306133557125141812009840009662733497578477395741589958741155007862285485649171111258286647871898412035813448185128487166238219335182872053769745063205146240398270221977832380760762866554366743397019522289256347615462644913261775369992728315584923236659323759817418582764754173499371387884058167010542953584434449476393697721676981883264752309900228411652423246081739021978704316749310333533596904537502580519003591630854375995694511316758712127072335981655643021189629703319518996608891858801563606731511756259150271536904664925444915995745598487882850973342179949112232261107451564475708164124601869338680457040736426834176357325238700023154772340405663484960868000544476177063934327405358840986142240740495891233632352852053087368646776262360895352822595554176491656178820976720387079767602962842304015276653872951276656719564661860009852322150747843167248021400524688931060413853949705429841350499311344844142812690878735649021359350878799892991941300536391836009746220081646980020619328507232729433224792490941993693654589654207336860144043824383616426523896328586666972201974975363869745131277430423497779482704923699635266814730743056122797451467295167944104959148848306171227734538923653674351260090426832081683750824578884795592847739029407231100114031692028834847718052811109661505435074338037197807509927683710026016782011198945921757041861903371723076024299552942686154078275262558274383125240246903963660244565495743790100779385689120612914314126748249032328644943967606168810945505133744051503793356677613465767506133403785838880077428117672171491305463631982985278470240463605873903007823368419732452411249428087806823995726037033029954428007284645945501678886834962638266386697203029172069359055069598160085557611071250819586513883262334808499877279404265739246453854314818930473784012514484630065265839504131463613716847280435096127167440453437234925013899740472517629852858007350702055473349166597916007035221009345839579731778913437555757452869569584725521765148238483120629334049015846611258643709781106104555540382601644817619693116271703781814763254333627569201647746337166728676209480537183667033744980348862985594703362234073685730010342405696049810927652018855284359782308568790335680407039194097771231043231125155243319999767116121609430970384357269423481232518264366416525210424503728896257581964154317685655227495297650147999531562443287526368243680227419154845562643905990854032891584723971919362819173221100539566110801807612543052603376782159573210538409554672405396295376724610561 / 466192705877572353389835231940601222951670101070134561239973049203355736797679164397624010041403280350403890172469013022072991611009891420460184365109844228257723354893359318552165960075193563963432495384489969443453032619043283947700320752061908507268402779707752507800584175761024396184554311926902174604278990716347817088215671830701294237737416084870577308225709433802172004179436883634224430186712907011255416169182951793876196020581124124871790191131026190817826295390668469484921153061628957182315532723627541561158527229962601975323545963581536503234088278520594697564268665056763039200953239157471488828427155355325822506814512931380692989201738194683671358027936731158164868428465160293810220914942856829006898677125278247051066719835366903281173005006414793603140852470513412338483125238870107400749975506910479167617492429188364161622090075380794841274064530078229588882401307373157838975174030799988659511414398333740739995186397781300193196193202757302833310640180721583180401458815210422678535674359307400703540019934939449079991546032209790021358416751689023180874723355242538833041051750335417189450271841435425842357705548951520412541807195011790313725815748442413653722842379292155259290874590674053279873682477462608416130078159361725049569136018863847769692711434304284648779994670136133745537730928927221014707549796285465899515205737181699779348683309919359212078797505708653203063359044435493651560670169020403706390659381796376355011534232215252329581727323206390983456393841689668653574596495767581620929318398055395068946540290994175198141078405809664168143560635261493106792900807310948239120108625189411525965104535724046516823181955565824022550341414576665251862252517262971555570824456334217133885793566352181827821799733460800562650876183474302696913558747118873383788058316498534666549582395993896633701984857946104957599858399846326694339088056360491291651580401291235916846039420291797011958951903525756596083400499833765015610160614682572190562581996175482897349716398200856207446583669457688398879025589194587622790093420738922616627264007078524712707438700538087126382407147484598708878421359007944754077319874892451013305146739264196759332202156279663574573700221915659282202010270420525069461223193977455730690186787162810767959480118208312050676001116374540365682589708346162179047550107133580021937933978889261168248964550472603798305071959551200598606625251939526329942481081494978482611041645840498095348015739875843098411344378730289107163926051684865403472243934262440308576997235702810712358963888128180945434447661046236283823959416606140230636235659383407828994007185098833311796358669296501389331388587322635430609807372093200683593750000000000000000000000000000000000000000000000000000 which has 137 decimals equal to 0.

[@tago3860]

@@Восьмияче́йник christ wow

[@LordSwordbreak]

@@Восьмияче́йник holy mother of Gauss

[@Pilchard123]

I've been trying to wrap my head around convolutions forever, so seeing that you're going to be doing a video about them has just made my day :)

[@3blue1brown]

Anything specific you're hoping to learn? Or any specific contexts where you saw them and were confused?

[@fygarOnTheRun]

Me too! I was always confused how convolutions seems to be meaning different things at once, like folding and multiplying functions and doing f(g(x)) ..

[@AwkwardDemon]

@@3blue1brown As a chemical engineer, the only context I've learned them in is just for how to use them to take inverse integral transforms (basically just using the definition of convolution). I'd love to see more about the motivation and intuition behind that definition

[@danieljensen2626]

@@fygarOnTheRun f(g(x)) is composition, not convolution.

[@Pb-rx1lh]

@@3blue1brown I find this topic in my statistics classes..i would like if u cover this in context of convolution of probability distributions.

[@harrywang2566]

So if we alter the series with 1, 1/2, 1/4, 1/8, 1/16… the integral will always be pi since the sum of this series will always be less than 2

[@agiri891]

But we would have to multiply by 2cosx

[@kaanetsu1623]

How can we check this if it is true or not

[@jacobhebbel6448]

@@kaanetsu1623 I could probably do it rn but I'm busy; but just use a calculator/desmos no? If not desmos use a graphing engine and input the function

[@TylerStillPlayz355]

That's not the same as the series from before, because the numbers were all decreasing by -1/2, your suggesting to do 1/2^n

[@thelightmare]

@@kaanetsu1623 one could study the convergence of the series towards PI

[@Scott-wd1cq]

These video's are so incredibly well made that, not only is the math beautiful and well-explained, but the scripts 3Blue1Brown uses in these videos is just as beautiful and meticulously constructed. This is one of those subtle things I love about science and math - that it teaches you to speak carefully such that what you say has exactly one meaning. It's a truly difficult art to master but if achieved, the speaker is effortlessly satisfying to listen to.

[@lemonlordminecraft]

Hey 3B1B team and especially Mr Sanderson, I just wanted to say your videos never fail to enthrall and impress me. You have such a way of communicating high-level concepts that makes me feel exceptionally well-informed about the subject matter you cover. As of 3 days ago, I've finished my Bachelor of Mathematics degree, 4 years after having my love of mathematics reinforced by your popular video about 4 points on a sphere. Your channel and its content are so important for young, mathematically-interested people and I cannot express how grateful I am for this content. In so many words, thank you.

[@3blue1brown]

Congratulations on the Bachelors, that's outstanding! And thanks for such kind words, it means a lot to me.

[@TheOneAndOnlyZelenkaGuru]

I think your space bar might be broken

[@therealsemechki]

@@TheOneAndOnlyZelenkaGuru utala pi toki ala (impossible, 100% fail)

[@kpw84u2]

@@TheOneAndOnlyZelenkaGuru it's probably just a moving averages problem 🤷🏽‍♂️ 🤣😂🤣

[@notsanger]

@@TheOneAndOnlyZelenkaGuru i think it might come from using a foreign language keyboard on an iphone. when i was learning Mandarin, if i typed an English sentence the space bar would add extra large spaces for some words.

[@johanneseller1081]

As someone who worked extensively with convolutions and Fourier Transforms in physics and engineering: This is a beautiful video and I’m excited to see where it leads us.

[@Herdatec]

Once he showed that moving average it made click in my head and all the lost knowledge about fourier and convolutions from my university came back to me.

[@mnguyen313]

@@Herdatec I flashed back instantly to second year college and getting a B- in Signals and Systems. Heard him say sinc(x) and everything repressed came back

[@Kyurem_originale_Form]

As an electrical engineer student as soon as I saw sinc(x) I immediately thought: Ah yes, definitely something with Fourier Transformation later in this video. Here we go again!

[@allanjmcpherson]

Yep! This is the foundation of all signal processing! Takes me back to my analog systems and signals class!

[@lyrimetacurl0]

😒

[@sebagomez4647]

Im also an electrical engineer student and we see this next semester. afterso many classes I realize the entire world can be broken down into vectors and sin() cos().

[@valinhorn42]

@@sebagomez4647 It's even cooler than that. Using the trigonometric functions is convenient because you're familiar with them already, and they're easy to generate with analog circuits. The Laplace transform and Z transform generalize this further to also take complex arguments (instead of a real number x). And in digital signal processing, all hell breaks loose -- Why not transform any function using a rectangular wave? Why not transform them using quantized waves? Look up leaflet transforms [correction: WAVElet transform].

[@jasonreed7522]

Systems and signals is the class that makes you appreciate Fourier and Laplace Transforms, and math in frequency domain / complex numbers as litteral magic. The trick is that litterally any real world function, and many "mathland" function can always have their Fourier transform taken and be expressed as an infinite sum of sinusoids or complex exponentials (which are easier to work with), and then you just do regular multiplication and perform the inverse transform and you have the answer. One of our jokes is that "laplace is god" because its just that much easier for solving differential equations. (And most high level physics equations are differential equations in their most generalized form)

[@johnreid4830]

I’m currently studying maths at undergrad level, and the difference between 3B1B and the teaching I am receiving is day and night. You do so much to motivate and illuminate with these videos. I know that to learn the detail will involve a lot of hard work, and then I’ll have to develop my understanding by exercises and problem solving. However, now that I am fascinated and have a picture, this is a joy, not a chore. Thank you so much and keep doing this sort of thing.

[@stephenbeck7222]

Yes and Grant has made what, about 5-10 hours total of videos in this manner for his channel? While in your math classes, you get 40 hours or so of content for every course. KZbin will always win out for ‘most interesting’ content. A good in-person educator will take the best of what is online though and bake that into the daily teaching.

[@cadthunkin]

Maths pronouns: they,them lol

[@MathOSX]

Excellent ! That may also be an example of why proofs by induction are required : observing the first terms of a sequence never tells you for sure what happens next ...

[@thecanmanification]

i’m currently studying electronic engineering and i’m pretty familiar with all of this frequency domain stuff, but the sudden “aha” moment I had at the end was really something else. 3B1B really knows how to neatly wrap together seemingly disparate pieces of information

[@trevorbradley3737]

I was taught by both Borwein brothers (Johnathan and Peter) at Simon Fraser University in math undergraduate here in British Columbia, Canada. Peter was a joy to take complex analysis with. Jonathan's 4th year real analysis course was... less joyful. Brilliant man, we as his students weren't ready to hold the volumous and requisite knowledge in our brains at all times. Still, I greatly appreciate the experience and am glad I passed his course!

[@trevorbradley3737]

The very best hour of my undergraduate was a day where Peter Borwein, 10 minutes into our scheduled hour long analysis class on a hot summer day, chatting about anything but the course material, said "I didn't want to teach today anyways", and we spend an hour just talking about mathematics and science. I would pay good money for a recording of that hour.

[@mastershooter64]

Woah nice! Was it analysis in R^n and general metric spaces or more like measure theory and functional analysis?

[@MaximBod123]

Had no idea they were professors at SFU! I've just started my first year at SFU as an undergraduate majoring in data science.

[@casperguo7177]

Very cool that they are teaching at SFU. SFU really deserves more credits than it gets. Despite all the trash talk from UBC, SFU seems to be quite strong in several departments.

[@casperguo7177]

@@MaximBod123 I am doing DS at Michigan. Seems like the SFU program has quite a bit of business focus that is absent in our program. Goes to show how underdefined the discipline is ig

[@faluffel]

Is it weird if I'm not studying or doing anything remotely to do with this kind of math, but absolutely loved it? It's strangely soothing and entertaining.

[@jaewok5G]

it's not weird. it's nice to find out that there are things you don't understand that will work themselves out in a very elegant way.

[@B0K1T0]

Not at all.. it's pretty much the story of my life :D Downside is I really have to put in some discipline to not be binging on interesting content too much :P (or maybe it is, but in that case I love to be weird)

[@kaijuno]

I find these so calming and beautiful, despite never really being good at maths. There’s such a sense of elegance and awe to these big concepts, and they always make me feel like I’m experiencing something beautiful.

[@pluspiping]

Not at all! Looking "under the hood" & getting an explanation of How Stuff Works is fun for the Curious, whether they're going into math/manufacturing/car repair/etc or not

[@cara-seyun]

yes

[@gerrero235]

I don't know if anyone will ever see this comment, but as an Electrical Engineering student, I guarantee that Fourrier and Convolution are very powerful tools. We can analyze an entire circuit through equations modeled using fourrier and laplace. Note: I was taken by surprise, I wasn't even looking for videos on this subject.

[@franciscofarias6385]

I'm not at all a math student, but I come to this channel every time I want to relive that feeling of "wow everything is connected, this is so beautiful"

[@bregottmannen2706]

Its so nice when you know enough math that you can figure out the problem yourself midway through the video

[@mccleod6235]

Yeah, god bless learning filter design many years ago...

[@me-me9ic]

Yeah must be, but that's not me lol

[@sypialnia_studio]

It's also not really nice when you don't know or better yet understand anything in the video from start to finish. That's me.

[@MattDiver]

A professor in college had this on his door along with a warning about assumptions and patterns. It's been in the back of my head for years to look into this and understand it!

[@SUNRA131]

I'm a retired machinist and I ran into this twice this while machining radii from for example 9.500" to 8.500" in decrements of .01". I called tech support and no one knew the answer to this. They had never heard of it. Now I know, 15 years later.

[@HemantKumar-xn8mn]

Could you please explain how Fourier transform comes into picture in your case ?

[@SUNRA131]

@@HemantKumar-xn8mn The radius I was machining decremented by .01" from 9.500" - 8.500". When the control got down to , for example, 9.130, 9.130-.01=9.12. Not so. the variable read 9.119999999... . Then when the control got down to say, 8.729999999..., instead it read 8.7299999999...8. This happened on every piece I machined. No one could explain why.

[@EinsteinsBarber]

@@SUNRA131 this is probably actually due to floating point imprecision and not due to the problem featured in this video. basically, theres only a finite amount of floating point values that can be represented with a certain number of bits, and since some numbers dont perfectly translate to a corresponding floating point value, itll choose the nearest one instead. most of the time this works fine, but sometimes it doesnt. a good example of this is if you try doing 0.1 + 0.2 in many programming languages, itll compute to 0.30000000000000004. entering 9.13-0.01 into python returns 9.120000000000001

[@SUNRA131]

@@EinsteinsBarber Makes sense. Thanks.

[@Bobbias]

@@SUNRA131 Going into it a little bit more: Computers represent everything as binary, including floating point numbers. The way floating point works is you divide your 32 bits (or 64, or 128... you get the picture) into 2 distinct parts. The exponent, and the mantissa. This is analogous to scientific notation in decimal: 1.2*10^5 has 2 parts: the exponent (10^5) and the mantissa (1.2). What ends up happening is that since we have a limited number of binary bits for both the exponent and mantissa, we end up with gaps where certain numbers cannot be represented exactly (without using more bits). In addition to that, it seems that nobody uses the error handling defined in the relevant standard to detect when a number is not representable. This can lead to compounding errors when an inaccurate representation happens in the middle of a multi-stage computation. If you're really curious, look up IEEE 754 on wikipedia :)

[@davidbaldock9321]

I was lost before he began.

[@christopherboisvert6902]

Keep going to learn calculus, I was a noob before and now I get it x)

[@somenygaard]

Why do we watch this stuff we don’t understand? I find it so interesting.

[@Richard0915]

@@somenygaardi watch it to absorb the information, not to really understand it, the information itself is just a collection of really interesting facts and they are fun to listen to lol

[@tamashellwig5275]

As a Hungarian-German, the name Borwein is pretty funny: Bor in Hungarian translates to wine, and so does Wein in German. So their name is basically wine-wine

[@deaconmaldonado7947]

Born to be a sommelier.

[@mortenbund1219]

And as just a German, you think to yourself why they're putting boron in that wine.

[@botanich]

​@@deaconmaldonado7947BORn to be a sommelier

[@apollyon1]

Arnold Blackback approves.

[@puppergump4117]

@@mortenbund1219All the other elements argon

[@Me-0063]

In a 20 minute video, 3b1b teaches what my school takes 1 month to teach

[@Alex-ud6zr]

I've always thought your visualizations are among the best I've ever seen. Thank you 3Blue1Brown for getting me back into Mathematics after graduating from university!

[@seneca983]

He uses a Python library called Manim to make them.

[@Kirby703]

@@seneca983 He created Manim! :)

[@ThePat123456ify]

As a math enthusiast that became engineer 25y ago 3B1B makes me feel I can still understand complex & fun stuff like this 😍 definitely the best youtube chanel ever, there was nothing like this before youtube

[@nva225]

Holy crap this is elegant, you have made scary maths approachable somehow. Well done

[@BernhardWeber-l5b]

Absolutely amazing. The problem itself and the quality of this video.

[@jajssblue]

These are the kind of awesome videos that I wish I had back in my undergrad in Physics. So helpful and intuitive!

[@ahmedhani9854]

For some reason watching this video the though of DNA telomeres jumped into my mind. The fact that they shorten but remain relatively functional all the way until that critical threshold after which they fail to produce coding sequence protection. It’s just fascinating how our world’s laws just mesh and meld into one another from math to biology to space-time geometry

[@FutureChaosTV]

Well, as one science communicator on youtube puts it: "Physics is everything"...

[@Asto508]

Only that there is absolutely no relation whatsoever between telomeres and this. It's an artificial relation that only exists in your head, sorry.

[@shottytheshotgun]

Thats not how telomeres work. They aren’t “used up” they are just buffers.

[@hell1942]

this person seems to know what they are doing, I know this is 3 months old, but hoe do you think they work

[@robbinxu6119]

This video takes me back to my senior-year signal processing class back in college and learning about Laplace transforms and convolutions. I knew that the term "convolutions" sounded familiar and it seems like Fourier transforms are just a special case of Laplace transforms! This is why I love your channel - it brings back memories of learning (and the trickiness of these topics) from the past and it's sending me into a deep rabbit hole of trying to remember much of this topic. I've never commented on any of your videos before but thank you for this great video and all the others you have done over the years.

[@platimus7984]

As an electrical engineering student, convolution and Fourier transform are very useful and interesting concepts. I loved this video.

[@calvinvlog768]

me watching these videos to feel smart, knowing full well that i don’t understand a word he’s saying

[@climbscience4813]

This is fantastic! I have actually used the relation between the convolutions of rect functions and the multiplied sinc functions in my work. The convolution of rect functions is actually one way to express a jerk-limited motion curve. Separating it into the sinc functions in frequency space can help tremendously to understand the impact that such a motion curve has on a control loop. Really cool to see this here! 🙂

[@hn396]

As soon as you started talking about rectangular pulses and the value of f(0) I immediately realized it was going to be Fourier frequency analysis and the DC offset, amazing video!

[@EvanPederson]

This type of stuff takes me back 20 years to my college days in the best possible way. Thanks for helping keep that feeling of wonder and amazement alive.

[@nbooth]

I'm paused at 0:34 and already my mind is blown. How is this possible?!?

[@BrownHairL]

These videos are just gorgeous. You make seemingly complex problems almost unnecessarily intuitive. It's a thing of beauty.

[@johnchessant3012]

Fun fact (also somewhat connected to the Fourier transform): You can actually integrate sin(x)/x using the Feynman trick of introducing a new parameter and then differentiating under the integral sign, but to do this you needed to somehow come up with the crazy idea of setting F(a) = integral of sin(x)/x * e^(-ax) dx from -infinity to infinity. (The rest is a routine calculation of finding F'(a), and integrating it back to get F(a), and substituting a = 0.)

[@hOREP245]

For what it is worth, what you are doing is essentially the Laplace transform. You just first note that the integral is even, so you only worry about the positive half of the axis.

[@HilbertXVI]

It's not too crazy once you realize you're just trying to eliminate the pesky x in the denominator

[@jesselin2935]

Guessing exp(-ax) is not crazy necessarily. It is done often in physics, because 1/x diverges when integrated, so one uses a strongly decaying function like exp(-ax) to "help" make it converge faster, then you remove the "help" at the end. A similar trick is used in quantum electrodynamics where the Coulomb force has a potential V(x) ~ 1/x. The exp(-ax) factor corresponds to if the photon actually had mass a, and then at the end of the calculation we set a = 0 because photons are actually massless.

[@tylerkrusemark9191]

Yep, you're definitely describing the laplace transform and it is actually a generalization of the fourier transform

[@kindlin]

I'm only just beginning to actually intuitively grasp the Fourier transform over the last year or so of some really excellent videos coming out, but now I got the Laplace Transform to try and figure out! lol

[@maximcerta1641]

he makes such niche and complex subjects seem so simple! very nice

[@kyspace1024]

And he's so cultured that he didn't use the word niche, he said "esoteric".

[@05degrees]

They aren’t that niche and complex but yeah! Master work. ⚙🕰

[@firebanner6424]

@@05degrees they absolutely are lol. Most people never even go past solving a triangle. For most people even basic differential calculus is completely foreign

[@NAG3V]

These topics are certainly ... complex

[@ikocheratcr]

Fourier transform and convolutions are not exactly niche ... It is bread an butter in electronics, to name a few: control theory, communications, signal processing.

[@lionel3921]

I am French chemist...very far from math in general... but your way of explaining and showing interesting mathematical things made me read my old book of mathematical analysis :D Thank you and please continue!

[@TheBartgry]

I get you. I study molecular immunology, far from the math lands too, but these videos help me grasp the wonderful elegance of mathmatical problemsolving. Fascinating stuff!

[@zixuanzhao6043]

Whenever I want my mind blown I came to this channel, works every time .

[@chiefsofnobles]

this guy is the bob ross of math

[@AaronPM55]

Fun to watch after finishing an electrical engineering degree. Feels like the second you found moving averages, I could see the convolution and Fourier transform. Made me feel like I learned something in the past 4 years

[@abebuckingham8198]

If this didn't immediately trigger you Fourier transform reflex as an electrical engineer you would have grounds to sue whatever school gave you the degree. That sort of negligence would be unheard of.

[@kasperholck5928]

Currently studying 'Signals and Systems' so a video about convolution is absolutely godsent. Great video as always!

[@shobhitmaheshwari8676]

I just finished a signal processing course and this is what we did all semester. So satisfying to have it explained here!!

[@michael0803aa]

Love this video! As an electrical engineering student, I immediately link everything together as soon as I see the rect(x) and the moving average!

[@Sauspreme]

Wow, I've never even taken a calculous class and somehow I was still able to fully understand why the pattern doesn't hold because of your explanation and animations! This is such a well put together video!

[@randomz5890]

Just when I was looking for a good maths problem to ponder, you swoop in to save the day! Thank you Grant for everything you do.

[@ColemanMulkerin]

Convolutions and Fourier Transforms were my favorite parts of math. Great introduction here.

[@vladudrea9747]

Always makes my day better when I see that 3B1B uploads

[@holyfool343]

My background is in optics rather than maths; as soon as you switched the discussion to rect functions, I could see the whole remainder of the discussion laid out. Very satisfying, and a great discussion!

[@paulhowell3514]

As soon as you defined the rect function I was thinking to myself "he's going to talk about convolutions in this video!"

[@theseusswore]

that little drawing sequence caught me by surprise, its really fun and i love how seamlessly it mixes into the video! didn't even feel like its a new thing, nice :)

I knew a lot about that, and still I learned a lot. Such a magical video. Keep this up. Can't wait for the next in the series.

[@dfunited1]

This takes me back to my childhood. Sitting in the Oakland Public Library reading a book about the unknown formula of an egg. You have such an amazing way of presenting!

[@caseyj9]

Please elaborate on the formula of the egg 🎉

[@dfunited1]

@@caseyj9 If I remember correctly, the formula didn't exist. There were close approximations. The book as a whole explored lots of mathematical mysteries both solved an unsolved.

[@frankjohnson123]

You gave me so much nostalgia with that comment! I loved trying hopelessly to understand math and physics problems as a kid but still getting exposed to some interesting ideas.

[@jittichaijuntaratin5303]

Thanks!

[@chinmayygarg]

I've always enjoyed your videos, but for some reason, I just think that this is one of your best videos. Beautiful animations and explanation.

[@HAL-oj4jb]

Oh man, can't wait for the next video! Even though I know convolutions very well and have used them quite often, I never was able to really wrap my head around how they work, and I'm looking forward to changing that with the follow up video :)

[@RPEndeavors]

Gosh. Convolutions were always a difficult topic to learn as it is tricky to wrap my head around computing them. The representation of the moving average to explain convulsions is quite elegant and cool. It is always great to understand the intuition and the deeper meaning behind math concepts. Can't wait for the next video!

[@misterdubity3073]

I'm familiar with all the concepts of this video except for convolutions, so I'm looking forward to the next one. Excellent, well thought out explanations and graphical representations.

[@DuarteMolha]

Your illustration at 12:40 showing how the initial wave form and the fourier transform can be calculated from it as a continous integral of the corresponding constituent waves is absolutely genious. For a person that thinks much more visualy than most this was a perfect for me Thank you

[@Anna.Bystrik]

👏👏👏👍👍👍 this is *the* best channel of its kind, the team never compromises the rigor while maintaining uncluttered vivid visualization! Extreme quality of their work, the modesty of the presentation, the simple fact the text and the formulae are correct and proof-read to near perfection (in contrast with their ubiquitous competition) , all these features make the channel uniquely useful in their contribution to noosphere :) 👍👏🏆

[@tanchienhao]

Amazing to see the convolution and Fourier relationship conspire to create this interesting pattern!

[@dr_workaholic]

Imagine testing a computer program for bugs and you find a bug in math

[@DEtchells]

I went through all this in college, but that was *many* years ago. It’s nice to see the beauty of it laid out again (and without having to worry about reducing it to practice on a test next week 😁)

[@gregnixon1296]

I agree. Having to learn something for a test takes the fun out of the thing that I'm trying to learn.

[@RLDacademyGATEeceAndAdvanced]

Excellent presentation. The way you have started the video, it compelled me to watch it, and I am happy to see that you have made the connection between Fourier transform and convolution process. The most interesting thing is that you have given clear justification why the integral value deviates from pi after the stage of 1/15. I am Professor and researcher in signal processing (more specifically Adaptive Signal Processing) and I tried to make extensive treatment to Fourier Series, Fourier Transform (FT), DFT, convolution in my channel. I hope if anyone is interested to learn more about these topics, he/she may follow the channel. I really like your animation. It is so beautiful and interesting. Thanks for your great effort.

[@scottmiller2591]

"The way I think I'll go about this is to show you a phenomenon that first looks completely unrelated but it shows a similar pattern" Signal processing engineer gets it at this point "It's just the convolution theorem, which is linear in the scalar multipliers and over addition." Functional analyst enters the room and observes "And this is true for not just the Fourier transform, but extends to all transforms that have the convolution property." Abstract algebraist enters the room and observes "And to all fields with a homomorphism where the scalars include the field of the rational numbers." Category theoretician enters the room, draws a triangle, and leaves.

[@miloszforman6270]

Lol. Unfortunately, I can't tell really tell if this is a realistic picture. What transforms are these, apart from Fourier and Laplace?

[@davidemiozzi8589]

I graduated in electrical engineering munching transforms and convolutions for breakfast, but never really understood what I was doing and where it all came from. These videos have been full of a-ha moments so far. Looking forward to the next one!

[@ntesla66]

Signals and Systems 101... Laplace transforms Heaviside step functions and convolution are our bread and butter as electrical engineers. I loved this video more than you could know!

[@jasonreed7522]

Electrical engineering is all about abusing complex numbers to cheat math and save time. The Fourier transform is probably my favorite function in all of math because of how powerful it is. (Along with the rest of the time->frequency domain transforms, we had a joke that laplace is god in college because it made differential equations so much easier, not to mention how it sinplifies every linear circuit into a simple transfer function multiplication and convert back to time domain, not that it was always easy to do so.)

[@Klarpimier]

4:53 Oh shit. Grant is a gamer

[@perbehaps]

The sheer timing of 3B1B releasing a video on the convolution theorem the same day that we started learning about convolutions in my signals class. Thanks Grant.

[@TechToppers]

I've had my own experience like that. I'm not any college student but yes sometimes some topics which I study really pop out on Grant's channel after some days which is a cool coincidence.

[@spdcrzy]

12:35-12:50 I have never seen a more beautiful and succinct depiction of WHAT Fourier transforms are, and how they work. Brilliant.

[@w.s6124]

As someone who never heard of Convolution and all that stuff. Things got confusing rather fast after like minute 10. But thanks to visuals and simplified explaination i could follow somehow. Top vid.

[@0SuperTacoMan0]

Just finished one of my EE semesters and we learned convolution and Fourier transforms/series. This video would of been so nice to see a few months ago. Still great to see though.

[@nirorit]

10:29 looks like an error, the negative powers are supposed to be below 1, not just the sum over an. (For example 3+5+7+9

[@chizhang2765]

a_n is the series of fractions. See 2:50 f.ex.

[@panostzo2713]

Why do you manage to blow my mind with every single video you make?

[@MohamedMahmoud-ey9tj]

Not only visuals are amazing, the wording script of what you say is insanely accurate and well-thought

[@donnerflieger3770]

I really like you continuing on Fourier transformation and convolution. Especially cause in my field of work it's not just an abstract math concept, but a real physical phenomenon - if you shoot a crystal with x-ray out the other side comes the Fourier transform of the crystal structure - inanimate crystals can perform Fourier analysis, not just computers and mathematicians, which I personally find really cool

[@rampadmanabhan4258]

This is one of the coolest things I've read today. Could you elaborate on how that happens by any chance, or point me to a source?

[@donnerflieger3770]

@@rampadmanabhan4258 Ok, basically it goes like this: Atoms in a crystal want to have the lowest amount of energy in their positioning to each other. So they will situate themselves a certain distance apart, everywhere. So we have the same distances, over and over, in a periodic structure everywhere in the crystal. If we now shoot an X-ray beam on that, the electrons of the atoms will to move in sync with the waves of the X-ray beam. This input in energy means, that they also produce X-ray waves of the same frequency, spherical in all directions. Because the crystal is periodic, this will all add up. But because they are a certain distance apart, the electrons are not in sync to each other, but to the highs and lows of the incoming X-ray wave. So if one wants to add them together, one has to account for this phase shift of 2pi * distance between any tow atoms / wavelength of the X-ray beam. So to get all outgoing X-rays one has to Sum over all atoms, or in the limit, take the integral over the electron density in the whole crystal, multiplied by e^(2*pi*i*distance/wavelength), which is the exact same formulation as the fourier transform. The distances between atoms get transformed into what we call the reciprocal space, similar as time resolved things get transformed in 1/time or frequency space. If you google "Laue method" you will see pictures of such experiments, where on the other side of the crystal you get the primary beam, and many different dots on the plate, going out of the crystal at a certain angle. Due to what is know as Braggs law, the reciprocal space is directly connected to the sine of the angle of the outgoing beam, and the intensity of each dot is proportional to the phase-accounted summation of the number of electrons in each so called diffraction plane. Because remember, the crystal is periodic? So the distances between atoms are too! And so each in plane of atoms each atom will have the same distance the next atom in the next identical plane, and so the amount of distances are finite. So these finite amount of distances mean that one gets a dot for each plane, and not the full plate full of varying intensity X-rays. So basically the crystal produces a fourier transform of itself. Or the X-rays do. Or both togehter. But one then has to do the step back, perform ones own Fourier transform to get the original atom positions, which is a whole other story. For more information I want to direct you to Kevin Cowtan, who has excellent resources on his Website of the University of York, especially "THe Fourier Picture Book" and the interactive tutorials for Structure Factors and Ewald Spheres

[@nileshkumar3090]

How can someone not fall in love with the beauty of mathematics ! You are brilliant in explanation 😘 love from india🇮🇳

[@WhiteHatMatt]

I like how he keeps saying “if you believe me” as if we don’t implicitly believe every word this man speaks while watching animated pi symbols.

[@Mirko88Fax]

Your videos are, by far, the BEST videos on whole YT... Explain these concepts, with the simplicity and naturalness you use... How it can be even possible? Out of this universe... Thank you 3B1B.

[@meurglysiii]

First I thought, that it is not very profound, since applying Fourier transform to this integral seems like an obvious thing to do. But then I tried reading the paper and I see that these mathematicians are studying something more general and it isn't that simple. Respect for making it seem easy.

[@MultiFunduk]

As a guy, who's already quite familiar with Fourier transform and convolution, and even with this "strange" fact of integral sequence breaking at 15, That was truly wonderful feeling of "holy moly, all that time it was just 1/3 +... + 1/15 > 1 and that explains everything" I just never gave myself a moment to ponder about why should it be so, Huge thanks once again!

[@Wakrar]

4:13 probably just something that you guys missed during editing, but I guess that integral of 2cos(x) shouldn't be evaluated as pi...

[@3blue1brown]

Right you are! Good catch. Consequence of defaulting to zero indexing :)

[@severaldata]

Mathematicians: Math isn't mathing

[@agentdarkboote]

This was thrilling and magical the whole way through. This is one of the few times I've been able to see what was coming at every turn - I just spent yesterday working on a problem involving convolutions, and when that little hint popped up about the relationship between the Fourier transform and the integral definition of a convolution, it was an exhilarating feeling!

[@JoseIgnacioZapata]

Wow those visualisations are magical, your channel never stops to give me goosebumps. Thank you so much

[@BrananPR]

My gut said this would be something to do with 15 being the first odd number with two distinct prime factors. I was expecting another video that was about the secret relationship between pi and primes. But this went in a very different direction and it was so incredibly fun to learn about

[@Wolforce]

oh man I'm reminded of the tons of exercises on convolutions i had to do for our waves courses in university. I really did like the science behind it, but the extensive math problems were a pain. Still, when we learn this in applied science and see what can be done with it in regards of waves, its fascinating.

[@jasonreed7522]

Convolution is when all the other electrical engineers in my class revealed that they had truly abysmal geometry and spacial reasoning skills, its fundamentally sliding graphs across eachother to set up a very ugly integral almost exactly like how he did the sliding average animation. Fortunately we quickly moved on to learning that you just transform to frequency domain, use normal multiplication, then transform the result back to time domain. (And make a computer do the actual arithmetic of it, systems and signals is the class you are told to make matlab do all of your integration for you)

[@BryndanMeyerholtTheRealDeal]

Bill Wurtz: ♪ How did this happen? ♫

[@georgwrede7715]

This left me breathless! Wow! What looked like an insane thing not only got explained, but even gave me the feeling that I understood (most of?) it! You are an amazing explainer of math!!!

[@johnny-chen1999]

Ah yes, the infinite loop of watching 3Blue1Brown videos because now I have to watch the Fourier video again to satisfied my curiosity. Then, another Theorem will surely show up which I will not remember perfectly, causing me to watch yet another video. Until, I have a complete understanding of every math used in the video, which is probably never, I will not finished learning. Eventually, what I seek to understand will return to this video again, therefore starting an entire loop again. So, I will just have to call it a day and cut the loop with a concept I have a "good enough" grasp on. Thus is my journey into the land of math.

[@cockroach7237]

I highly appreciate the math behind your videos. A video on Taylor's remainder theorem would complement his existing videos on Taylor series and enhance our intuition in calculus.

[@code_explorations]

The world does not deserve a video of this quality. Thank you so much, Grant.

[@divy-07]

Those visuals are top notch 😍😍, always astonishing how you can teach complex concepts in such a simple way.

[@thishandleistaken.]

You haven't finished the video yet.

[@der.Schtefan]

I mean, you're the gold standard for pretty much every knowledge KZbinr. I am an engineer and my brain was just blasted open, as it always is with your videos

[@arjunnanduri]

Gotta be honest. I dont know much about math as a Calc BC student in highschool but the fact is, watching something that is interesting and presented in a format that expresses an attitude of wanting to teach you something keeps me interested. Your voice also soothes me. If it was someone else's voice I would not care as much haha.