Complex integration, Cauchy and residue theorems | Essence of Complex Analysis #6

[@mathemaniac]

Unlock new career opportunities and become data fluent today! Use my link bit.ly/MathemaniacDCJan22 and check out the first chapter of any DataCamp course for FREE! Support the channel on Patreon: www.patreon.com/mathemaniac Merch: mathemaniac.myspreadshop.co.uk/ [hopefully, this time, the pinned comment won't disappear like last video, cos for some reason KZbin decides that a similar comment is spam lol] Please refer to the previous video for why I skipped the differentiation part: kzbin.info/www/bejne/fF7OiqisiN52Y6M This is the end of this video series, although I wouldn't say that complex analysis will disappear on this channel forever, just that it will only have occasional appearance if I feel like it. I know it's a bit "irresponsible" to leave out the details in the last bit of the video, but the main point in that part is that residue theorem is useful in real integrals, and to be honest, videos on this channel are sort of "inspirational" rather than "educational", in the sense that it is not intended to be rigorous, as said in about 15:26. If those details are really in demand, I could make a video about it, but it will most likely on the 2nd channel: kzbin.info/door/WdGv5veEBYCn99pT7XJsjwvideos

[@jacobhoward7579]

Oh my God dude... I have been literally waiting for YEARS for a KZbinr to make a professional visualization of this!! ... Words can't express my thanks... 🙏

[@dnickaroo3574]

I have fond memories of Complex Analysis from the 1970’s, but have only returned to Maths since retiring. I have studied Feynman on Theory of Fields recently, so I love the connection. Thanks for providing this outline.

[@nosnibor800]

I watched this out of nostalgia. I am a retired Electrical Engineer and back in 1975 I covered this work on my final year maths syllabus of the HND, at Manchester Polytechnic, UK. I still don't fully understand it ; maths is so beautiful. I think this is applied in advanced control systems.

[@RAyLV17]

Yes it is applied in control systems. We n need complex visualization to determine the stability of a system. Something like poles and holes. I don't remember either haha, I learned it in my bachelor's in mechanical engineering.

[@nosnibor800]

@@RAyLV17 Ah the crossover between mechanical and electrical :-)

[@naman4067]

@@nosnibor800 now we have internet we can make our knowledge more deeper. Easy life 😌

[@rahulpsharma]

I think used in Nyquist Criterion

[@douglasortner6255]

Im a freshman in college for Electrical Engineering, looking forward to the challenge of figuring this stuff out.

[@DrTrefor]

Very impressed with this complex analysis series, well done!

[@mathemaniac]

Thank you so much :)

[@duckymomo7935]

I agree, it was so clear, beautiful and insightful 3B1B quality

[@naman4067]

Hi trefor

[@scraps7624]

Your videos are incredible as well!

[@zarifhasan1696]

You should make some videos on this topic too😁

[@samyaksheersh712]

Why on earth wouldn't anyone watch this till the end? It's such a beautiful result so elegantly presented! Keep up the good work!

[@mathemaniac]

Thanks for the compliment! Although I think that's all well-meaning, I do think that asking 40 minute of someone's time is difficult, however good the content is.

[@samyaksheersh712]

@@mathemaniac Yeah, Being a high school student, I didn't consider that. But it was a nice video anyway, and the results were really beautiful

[@moreasmorebaes9996]

@@samyaksheersh712 a fellow highschool student hear, how did you clear the prerequisites for complex analysis? I have to prepare a presentation and I really wanna make it about the value of residue calculus to solve real integrals and I barely know single vari calc.

[@neplatnyudaj110]

Flashbacks dude, flashbacks. But on the other hand, there were worse things in control theory classes.

[@nezavipavc172]

Physics student here, and I have to say that this has made my understanding of complex analysis *so* much better! You explained everything way more intuitively than how our professor presented the material, so thank you for making this video!

[@James-bv4nu]

The complex realm is so unlike the real realm, one can easily be overwhelmed and get lost without a tour guide. Cauchy's work provides the tools and perspective with which one at least has a fighting chance to tackle complex problems. Your video very nicely ties together Cauchy's theorems, like walking us through Cauchy's thought process. Well done, and thank you.

[@JackDespero]

What i love about watching videos of "basic topics" that I already know, is that each person gives it a slightly different perspective, and I always learn a new nudget of knowledge, a new way to visualize or understand something. Thank you for the video.

[@evaxu1325]

40:08 Watched to the end! An Oxford math student here :)) I am so grateful that you made this series of video. I was previewing complex analysis during the summer and suffering until I discovered your essence of complex analysis. It made the subject much less daunting and helped me a lot during the term! It even cultivates in me a love for complex analysis! There are not a lot of intuitive videos on university math like what you did. I was so excited when I found your channel and I recommended these video to all of my friends after the term has started :) Genuinely thank you so much! ❤️

[@MaxxTosh]

By far my favorite video covering complex analysis, ever. You put together nearly all the integration topics so well!

[@mathemaniac]

Glad to hear it!

[@abdullahalsakka]

You are an amazing creator and I don't know how to thank you enough. I am a physics engineering student and I have been struggling so long to understand the intuition behind complex integrations. I can do them but unfortunately by memorizing how to do them, I was blown when I saw this simple intuitive video and I really want to thank you deep from my heart. Also as a side note, I really liked how you explained the complex integration using physical quantities like work and flux. Keep up the great work and I hope soon enough many people find this channel and explore the fun and intuitive sides of mathematics.

[@sudhirfadnavis9312]

Thanks. I am electrical engineer studied engineering 4 decades back. This video seamlessly put all the maths concepts together.

[@nicholasbohlsen8442]

22:35 I love seeing the multipole fields of electrodynamics appear as the Polya vector fields of the inverse powers. Its just such a cute connection between the multipole expansion and laurent series

[@mathemaniac]

Yes indeed! Though obviously in "the real world", we wouldn't see this connection very well, because we live in 3D!

[@lifehope4201]

@@mathemaniac Did you imagined everything before making this video or you visualised everything using software first .......?

[@samuelmarquardt7884]

I love that the ads are between segments instead of annoying me while he's explaining

[@jowillll]

The first time I watched this video was during a trip to Bali around 2 years ago, I was still in middle school then. I am now in university and I have repeatedly came back to this video to check my understanding of complex analysis. But among those countless times, this time after I've partially studied the basics of complex analysis at university, I can say I have finally understood everything said in this video. I think it's been a really interesting journey, and I hope someone out there going through the same journey I did can find this comment as motivation. Don't worry if you don't understand everything said in this video now, but don't give up! Keep gathering more knowledge and keep being curious! Cheers :)

[@068LAICEPS]

I love everything related to Cauchy's mathematical topics. Now I love this Channel.

[@johnchessant3012]

Amazing! Really appreciate the effort you put into these videos, of course I watched to the end :) To add the details for the integral: The magnitude of the integrand exp(iz)/(z^2+1) on the semicircle of radius R is bounded by 1/(R^2-1) while the length of the semicircle is πR. So the integral on the semicircle is O(1/R), which goes to 0 as R -> infinity. The residue at z=i of the integrand is the limit of (z-i)exp(iz)/(z^2+1) as z -> i, which is exp(-1)/(2i). So the integral is 2πi times that, which is π/e. A lovely answer!

[@mathemaniac]

Thanks for providing the details here!

[@josipfleis9223]

Back in the '80-es, I managed to solve those integrals, using this method of complex integration (I use a book emineter.files.wordpress.com/2016/03/kompleksna-analiza.pdf) Although I was quite "well trained" in this field of mathematics, I get fully understand this watching Your video!

[@markseidel19]

Hands down the best complex analysis video I have seen on KZbin. Concise, conceptual, and each topic relates to the previous one clearly.

[@ativjoshi1049]

This single video feels like a crash course on Complex Analysis

[@mathemaniac]

Yes, indeed that's the intention.

[@ToriKo_]

Ur comment piqued my interest

[@oldsachem]

Best thesis explanation of calc stated in the first minute I've ever heard.

[@yinq5384]

Great as always! I never saw the "Work + i Flux" interpretation before. It's really helpful, just like the area/mass interpretation in real integral.

[@mathemaniac]

Glad you enjoyed it!

[@nutelina]

@@mathemaniac Please continue this trend, it was a great help indeed. Math can be confusing because of all the details, I love how you keep it simple with great analogies and don't slip into side-quests like 1blue1brown.

[@BambinaSaldana]

Hey guys I have a question. So area, velocity, acceleration, etc., all have representations in calculus, right? Area/distance is represented by the integral, velocity is represented by the derivative, and acceleration and its derivatives are represented by higher order derivatives. So, what is the physical interpretation of the antiderivative? Say we have a function f(x). This represents our distance. What would its antiderivative be? Is the antiderivative just pure math that's only really useful for velocity and higher order derivatives, or does it have actual use in physics?

[@TheTKPizza]

@@BambinaSaldana Maybe a bit late, but since you were talking about time derivatives of spacial coordinates (velocity, acceleration, etc.), the time based "antiderivative" would be called "Absement". It is basically an overall measure of the distance and time, an object has been displaced from its initial position. Feel free to read the wikipedia article abuot it for more details. I think, it is a very smart question, I was wondering the same a few years ago and looked it up. :)

[@BambinaSaldana]

@TheTKPizza Ohh so the antiderivative is basically how much distance was covered? Or how much distance the object has had from its origin?

[@r.maaster7451]

I'm physicist and this video is really satisfying to watch. It summarizes the math course I had about complex analysis. Nice done!

[@kevincerda6636]

As someone who's never taken complex analysis, you make it so interesting and engaging to keep watching. I admit some lengthy videos can be intimidating and hard for me to understand, but something about this one made me watch it through the end. I really do feel like I learned a lot from it!

[@mathemaniac]

Really glad that you like the video!

[@markbogosian9436]

I have run a successful arbitrage firm for over 30 years. I am awed by how much there is left to learn. Thank you for your efforts.

[@vkv392]

As a physics student....this blew my mind .....no words to convey my regards..thankyou very much for your effort..I was smiling out of pleasure whole thile time while watching your explanation...😌😌😌

[@mathemaniac]

Thanks for the kind words!

[@ЕгорКут]

I am supposed to start studying complex analysis in nearly 1.5 years from now. But it's really nice to have some good intuitions before you fill in the details. Thanks for your video!

[@mathemaniac]

Glad it helps!

[@mastershooter64]

why wait 1.5 years when you can start now!

[@viktorgeorgiev2119]

Wow, I think it is impossible to describe all kind of complex integration in 1 semester, but you did it in one video. Congratulations!!

[@matta2389]

I have a complex analysis exam in a few days. I cannot begin to explain how grateful i am for this video's existence

[@betacenturion237]

Hey Mathmaniac, I'm taking a graduate level mathematical methods for physics class and we are currently going over this. I can't begin to explain how helpful this video has been in trying to understand the content of complex analysis and I wish I saw this sooner! Thank you!

[@TheKshayK]

Im an electrical engineering student and i have a test in this course in two days. You just gave me so much intuition about this course and this 40 min video is worth like my 3 months of studying!! Thank you so much great content and great video

[@happynightmare2332]

theres something about watching a 40 odd minute video about some maths concept i have like no idea about and picking up bits here and there which i already understand and piecing it together which just gives me joy in life

[@cavesalamander6308]

As you know, education is something that remains unforgotten after graduation. After 40 years, I only remember: 'residue', 'contour', 'the integral over the contour is equal to the sum of the residues'. Thank you for the video.

[@lapisdust]

Studied this 40 years ago as a Physics student, never had as intuitive an explanation as this! I've been looking at videos and through my books just to refresh my memory. This is the best so far of all in Complex Integration. Watched all the way through - was a little sad it ended, hope to find more of your work! Definitely subcsribing!

[@mathemaniac]

Thanks for the subscription!

[@mastershooter64]

40 mins??? the gods have blessed us with more amazing content bois!!

[@shinjirigged]

I love when we make sure that we get the small assumptions right. Thank you, and Bravo!

[@slippinchillin]

A course I barely passed, explained in half an hour Salute to this guy

[@michaelperrone3867]

I remember seeing this in a mathematics class in college but with your explanation, I finally understand and appreciate what's going on here. Thanks! You're excellent at explaining things.

[@mathemaniac]

Thanks so much for the appreciation!

[@michaelperrone3867]

@@mathemaniac By the way, what's the software used here? Is this the same thing 3Blue1Brown uses? I'd like to learn how to use it for maths visualization for my own projects too; you should add a link to it in the video description or maybe to a sponsorship with them if it's a company.

[@arttrust7629]

Really thank you 🙏, from India. I completed my M. Sc in mathematics. Your contents are helping me.

[@YourLocalCafe]

I am a highschooler so this went way over my head, but i must say that you are doing a great job at making these and you must keep at it! Your quality and the quantity with that kind of quality are both spectacular. Your channel is heavily underrated, but hey i am here!

[@mathemaniac]

Glad you like them!

[@YourLocalCafe]

@@mathemaniac :>

[@YOUSIFPOTATOYT0]

This the one of the if not the best complex analysis explanations I've ever seen !, thank you for your efforts .

[@mathemaniac]

Glad it was helpful!

[@uwuifyingransomware]

I have an exam for complex analysis coming up, and happened to get this video recommended. I thought against watching it in my study breaks to avoid overworking myself, but I’m glad I watched it fully! There’s a lot of intuition here that nicely complements the proof things I’m studying. I also went into this knowing not much at all about vector fields, and your explanations of those made the video still possible to follow. Overall, brilliant video :)

[@RobotBlueprint]

I have watched this 40 minute video and am leaving a comment to make this fact evident to the author. Thank you for the video!

[@daigakunobaku273]

Your video is the best explanation of Cauchy's formula I've ever seen, and I've read this part in, like, three different textbooks. Please, continue with this series, it is damn good

[@mathemaniac]

Thanks for the compliment! However, as explained in the previous video (linked in the pinned comment), I would probably end this series, though it does not mean complex analysis will not appear on this channel, just maybe some occasional appearance rather than a series. Or to think of it this way, those topics are not "Essence of" anymore.

[@alejrandom6592]

Man, I thought I understood this but only now after taking 4 pages worth of notes I think I actually start to grasp the profoundity of this.

[@Miguelocod]

Its' 4 AM and I've just watched the whole 40 mins. I had complex analysis for electrical engineering course and it was the hardest, and I never understood anything of it, residues and countours just feeled magical. Until this video.

[@anguswu2685]

watched till the end! just learned it last term in second year electrical/computer engineering but it’s so refreshing to see it with graphics

[@carmelpule8493]

I followed this video to the end and it is a wonderful video, . I am an engineer and I see this contour integration as follows. If one is integrating (1/z)*dz and z is taken as z= e^j(theta) then 1/z= e^(-j)(theta) and dz= je^j(theta). d(theta) Basically these are vectors rotating in the clockwise and anticlockwise direction. If the numerator is rotating in the same direction as the denominator there will be no resulting rotation of (1/z) dz and the vector integral will be along a line. the same length as the circumference = 2 pi. . with the j included because of the derivative. If the numerator is not rotating at the same rate as the denominator then the final equation will rotate and so its integral is zero. It is all a beautiful engineering operation and in the case of (1/(z-a)) dz then at location a it is the only expression that does not rotate and all the other location will rotate because of dz rotating. when z= a +e^j(theta) . At location a the function (1/(z-a)) dz is not rotating but all the others a other " poles b,c, d etc. are rotating so their integral is equal to zero. In other cases there are the residues to care for. The same for (1/(z-a)) dz and (1/(z-a)^n) dz

[@riyannaffanusafara5940]

Come for Cauchy, left with "I am going to math olympiad". I might not understand the half of the video, yet, but I will harness the power of complex analysis to its fullest. I'm going to get a hundred 💯

[@acer1210]

I am a student of pdusu, sikar, and here we don't have any kind of knowledge about what an integration even means. We just memories the formula and that's it Thank you so much for the true knowledge of complex integration.

[@sinkingboat101]

This summarized almost everything we had in a semester of Complex Analysis for Engineers, in just 40 minutes and still quite comprehensible. Btw, I watched to the end

[@nathanisbored]

i watched to the end, i learned a lot of what i wanted to learn in this episode!!

[@mathemaniac]

Glad you like it!

[@nelsonmvillagra]

I really saw until the end of the video. It is great! I am an electronic engineer from Argentina. I loved the way you explain these advanced math topic. I am reviewing some of this concepts to analize control systems. Thank you very much! It is very helpful.

[@genevievebrooks9605]

Watched the whole thing, it helped me enourmously the week before my complex analysis final!

[@pendragon7600]

whew, new mathologer video and new mathemaniac video in the same day. excellent

[@mathemaniac]

Haha it just so happened that we both upload on the same day.

[@IvanGonzalez-rp2xr]

Amazing work! I've watched the Complex Analysis series from start to end. You truly showed the essence of complex analysis in the nost natural and intuitive setting, thus conveying its elegancy throughout the journey. From one math educator to another, I tip my hat to you!

[@voorhs]

thanks a lot! i used it to prepare for my test and on contrary to those boring theorems from textbook your visualization and insights are the precious drops of living water!

[@superj1e2z6]

The way you beautify the agonizing part of complex integration makes me wish this existed years ago

[@mathemaniac]

Glad it helps!

[@leothethrasher1100]

I watched the whole thing. I took complex analysis a couple years back and this video was a great review of what I learned back then! Thank you for the upload!

[@mathemaniac]

Glad you enjoyed it!

[@Elaine-y4c]

Thank you a lot for this brilliant video. I have finished the whole list. It very well presented and I can understand the elegancy of complex analysis as a beginner with no background knowledge.

[@navjotsingh2251]

Please go into more detail about the ending! You are doing the Lords work, the more videos you can make about complex analysis the more better humanity will be!

[@mathemaniac]

Thanks for the compliment! Though I would most likely not be uploading here even if I would go into the detail there - instead on my second channel.

[@navjotsingh2251]

@@mathemaniac either way, us students greatly appreciate any efforts, and I think you can become one of the best go to channels for complex analysis. It would be good if you could maybe in the future show how we can do computational complex analysis and maybe even show how complex analysis can be used to analyse the complex version of fourier transforms/fourier series. There are so many things that you can do to help us and make you become a golden channel!

[@nerkulec]

Amazing! Every second of this video was well worth it, will surely rewatch it again to internalize all those ideas. Thank you!

[@camilocagliolo]

Excellent! I watched till the end. I've already studied this in college but it always strikes me as some sort of trascendental knowledge I never get to truly, deeply understand. That every value of an holomorphic function is defined by all the other values of the function, that if one derivative exists then all of them exist and are so simply related, and that only the 1/z component or pole of a function has an integral different than zero... is absolute madness. Why would math be like that? Why is it so elegant and powerful? And I know the answer, it's because IT HAS to be like this. It's absolutely wonderful. Seriously, before college I was an atheist and I ended believing there must be some kind of God behind all this.

[@michaelhall6187]

Getting ready to take a course on complex analysis in the next semester and youtube recommended this video to me, very helpful and simply explained so I’m much more confident now about the class! Thanks for the great video 😄

[@joeydeacon1]

Thanks, love the connection to the Polya field, work and flux, helps to understand why integration over the poles of holomorphic and meromorphic functions gives the results they do, other than just using Cauchy's theorem or the residue theorem outright every time.

[@vahehawi9503]

Exactly what I needed in order to fully understand the concept... It's amazing !!

[@jainrohit0123]

Very clear and lucid explanations. Watched it till the end. Please continue for a part two for this series.

[@mathemaniac]

Thanks for the appreciation! However, as said in the previous video, linked in the pinned comment, this is the final video in the series (yes, there are videos before). But this is not to say that complex analysis will not appear on this channel, just that it will only have occasional appearance.

[@dashnarayana]

Great pleasure to watch and learn the old concepts in modern student friendly manner. Great research indeed!

[@CanYalnz]

You're absolutely amazing and it's a privilege to have the opportunity to watch your videos.

[@mathemaniac]

Thanks for the appreciation!

[@MattMcIrvin]

The residue theorem is still the closest thing to wizard magic I know of in mathematics. Breathtakingly powerful and almost too easy to apply, where it does apply. I've never stopped being amazed by it.

[@derickd6150]

Epic video 😁 glad you at least showed part of how it would be applied. I think a lot of what makes it so interesting is how you can solve a seeming unrelated integral like what you showed with cos there. I think seeing that is what will make people go look up more

[@RandomGuy-ie2cb]

Great! Just finished a rigorous Complex Analysis course and this video series provided me with a lot more intuition and down-to-earth examples :)

[@merich4940]

Good god, this video is one of, perhaps even the best video on complex integration.

[@samuelthecamel]

Recently I made a simple integration program on my graphing calculator, and one day I was curious to see if it could do complex numbers. Turns out, it can! The specific contour is just the straight line path between a and b, which means it can't do most of the stuff from this video. But, the fact that it works is surprising enough.

[@MasterHigure]

Complex numbers really aren't that much of a step up from the real numbers. Surprisingly many things just work when you lift them up.

[@alejrandom6592]

retaking my math hobby. this is beautiful.

[@asamenechbayissa553]

I didn't realise this was a 40min video until I finished watching it. I really like this series.

[@mathemaniac]

Thanks so much for the support!

[@viggosimonsen]

Very enlightening on the physical analogies of flux and work. Great presentation. Thanks

[@datamoon]

I watched all the way through and this was super helpful. I am working through a self study of complex analysis and the texts I am using do not make use of the vector fields at all like you do. Some very dry and abstract set theoretic stuff that is. This really helped me make intuitions which is exactly what I needed, so thank you!

[@tahmidkhan7854]

It was really fantastic. As a CSE student I have Complex Analysis on my course. Never thought about it. Thanks for this wonderful explanation.

[@lourencoentrudo]

You've outdone yourself. Superbly presented!

[@mathemaniac]

Thanks so much!

[@TVTGLive]

Incredibly well done, please keep continuining this series.

[@mathemaniac]

Stated in the pinned comment that this series is ending - though it does not mean complex analysis will not appear on this channel - just only occasional appearance.

[@fattimiv]

Still here! Complex analysis was my favourite topic during my degree and it made me smile to hear it all again. I really like your videos, by the way. I think they are very well animated and explained!

[@mathemaniac]

Awesome, thank you!

[@vigilantradiance]

I've watched this video all the way through like three times, love your presentation of the material.

[@butchtoughman2376]

This is absolutely phenomenal, thank you so much for your work!

[@idokein2]

As someone who teaches this topic, this is amazingly put together and very well explained. And yes I watched all the way to the end

[@mathemaniac]

Thanks for your support!

[@LorenzoWTartari]

Watched the whole video, it really helped a lot, i just finished taking an introductory complex analysis class this semester, and had gotten a pretty good grasp of the key concepts, but i still managed to find a few key insights that hadn't dawned upon me while taking the class, and that will definitely stick with me

[@mathemaniac]

Glad it helps!

[@Boringpenguin]

Complex integration was one of the section in complex analysis that literally blew my mind when I first learned it in college, and I can tell you that my mind is still being blown away after watching this video! This is such a concise yet intuitive take on some of the most beautiful results in complex analysis, possibly even in the whole of mathematics. Very well done :)

[@mathemaniac]

Thank you so much!

[@douglasstrother6584]

You're pretty sneaky, Mathemaniac, giving us a cliffhanger like this! Looking forward to a Part 2 sometime soon. Really nice presentation.

[@黃彥淳-j4q]

Even without finish the integration at the end, this video still really helpful. Thank you.

[@RogerZoul]

I watched to the end. I very much enjoyed it. I teach this material in one of my graduate courses for advanced electromagnetics. I really appreciated your graphics! That really makes it a lot easier to visualize these concepts. I have been teaching without the aid of such for a long time. Oh, the residue theory is the most important part of this for what we do (its mostly math but aimed at engineering). Subbed!

[@mathemaniac]

Thanks for the appreciation!

[@yfidalv]

I watched till the end :) This video made me even more excited for when I’ll take complex analysis in college, really enjoyable!

[@mathemaniac]

Thanks so much! Not many people will watch till the end!

[@Noemi.39]

Yours are some very rare and precious video, i admire you.

[@mathemaniac]

Thanks so much!

[@sdsa007]

Of course I watched the whole thing ! Lovely reference to Feynman ! Very Beautiful as always ! Thank you for your math inspiration on this beautiful day!

[@jacobhoward7579]

I've been waiting for this video for ages... I do not have the time to watch it at the moment but I wanted to comment immediately for the algorithm... Again, if you've read my personal comment to you... Thank you ... Thank you again 🙏

[@mathemaniac]

Hope that you will enjoy it!

[@joaofrancisco8864]

What a great video! This channel only gets better and better

[@mathemaniac]

Thank you so much!

[@xavierlemaire4443]

I stayed till the end ! I thank you so much for this video because I have a homework to give about residue and I really needed some other look on complexe analysis. I think I'll watch again

[@ez5201]

One of the best videos I've watched recently. So brilliant! A huge thank you for such a good work!

[@KOMATH]

I love every bit of this video! Absolutely amazing!

[@stevenwang7520]

Thank you sir, this is the most intuitive explanation i have ever seen, and i truly hope there will be a follow up video on this topic.

[@HaramGuys]

Wow this is such a great insight, feels like complex analysis translated into electrostatics and borrowing the physical intuition from it

[@tox1678]

Thanks! I was looking for an intuitive explanation on this topic. It helps a lot.

[@chessematics]

39:56 i finished this whole playlist in one sitting, so i guess you're not the only one questioning sanity