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

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!
I can't pronounce "parametrisation" lol
A crash course in complex analysis - basically everything leading up to the Residue theorem. This is a more intuitive explanation of complex integration using Pólya vector field. As is the case for all videos in the series, this is from Tristan Needham's book "Visual Complex Analysis".
You might notice that my explanation on parametrisation is a bit similar to the Jacobian, and you will be right! Jacobian is really important in this area (and also understanding complex differentiation and Cauchy-Riemann equations).
I have made this slower in comparison with some of my other videos, because when I myself watched some of my other videos that are faster, I couldn't comprehend if I was not paying too much attention on the screen, let alone the audience watching it for the first time. If somehow, miraculously, you think this is way too slow, feel free to speed it up!
I said the more general Cauchy integral formula is related, because in my original plan, I did want to say that Laurent coefficients take on exactly the same form, but it just occurred while I was finally editing the video that we don't find Laurent coefficients using integrals, and I don't want to send my Cauchy integral formula bit to waste, so here it is.
Throughout this video series, of course I have left out lots of theorems in complex analysis, only talking about the things that I find more "applicable" (read: more audience want to watch). Things like Fundamental theorem of algebra, or maximum modulus principle, or even winding numbers are not presented, but in my defense, they are not really "essence of" anymore, because they use the concepts that we have developed in this series instead - like Cauchy integral formula as seen here.
This video was sponsored by DataCamp.
📖📖MORE READING📖📖
(Simply connected version) en.wikipedia.org/wiki/Cauchy%...
(Homologous to 0 version)
www-users.cse.umn.edu/~brubak...
Surely, You’re Joking, Mr Feynman: www.amazon.co.uk/Surely-Youre...
Math StackExchange thread: math.stackexchange.com/questi...
Residue theorem example links:
Wikipedia: en.wikipedia.org/wiki/Residue...
[The example that I nicked from]
You will need - estimation lemma (a.k.a. ML lemma): en.wikipedia.org/wiki/Estimat...
If you want to watch other videos on the exact same integral instead (although I think the Wikipedia page is a more “elementary” way of finding residues), you might want to have a look at:
Classifying singularities: sites.oxy.edu/ron/math/312/14... (page 1)
More ways of calculating residues: en.wikipedia.org/wiki/Residue...
KZbin videos talking about exactly the same integral (though they all assume quite a bit of familiarity of the above):
Flammable maths: • A beautiful result in ...
qncubed3: • Complex Analysis: Inte...
tahiri math science online teaching: • Integral of cos(x)/1+x...
🎶🎶Music used🎶🎶
Aakash Gandhi - Heavenly / Kiss the Sky / Lifting Dreams / White River
Asher Fulero - The Closing of Summer
Video chapter:
00:00 Complex integration (first try)
06:01 Pólya vector field
08:18 Complex integration (second try)
12:27 Cauchy's theorem
18:39 Integrating 1/z
22:28 Other powers of z
28:26 Cauchy integral formula
31:43 Residue theorem
36:14 But why?
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
Social media:
Facebook: / mathemaniacyt
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Twitter: / mathemaniacyt
Patreon: / mathemaniac (support if you want to and can afford to!)
Merch: mathemaniac.myspreadshop.co.uk
For my contact email, check my About page on a PC.
See you next time!

[@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!

[@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!

[@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?

[@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!

[@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! ❤️

[@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 .......?

[@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

[@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.

[@068LAICEPS]

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

[@sudhirfadnavis9312]

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

[@samuelmarquardt7884]

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

[@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.

[@nathanisbored]

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

[@mathemaniac]

Glad you like it!

[@user-ls8fw6zm3j]

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!

[@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!

[@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!

[@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 💯

[@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!

[@nerkulec]

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

[@shinjirigged]

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

[@oldsachem]

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

[@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!

[@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 :>

[@vahehawi9503]

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

[@butchtoughman2376]

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

[@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.

[@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

[@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!

[@genevievebrooks9605]

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

[@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.

[@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!

[@KOMATH]

I love every bit of this video! Absolutely amazing!

[@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

[@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!

[@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.

[@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!

[@ez5201]

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

[@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!

[@mastershooter64]

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

[@vigilantradiance]

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

[@CanYalnz]

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

[@mathemaniac]

Thanks for the appreciation!

[@juanfranciscocanesesmarin7086]

Watched the whole thing! awesome video on complex analysis

[@sissie6843]

Your explanation is sooo nice. Thank you!

[@dashnarayana]

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

[@adriencances134]

As usual, this is fantastic work!

[@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 :)

[@user-sm9jt9oc5r]

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.

[@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.

[@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

[@BrianAmedee]

Such high quality learning materials for free, thanks for making this!

[@mathemaniac]

Glad it was helpful!

[@alejrandom6592]

retaking my math hobby. this is beautiful.

[@PunmasterSTP]

I'm here and watched to the end, so another part is definitely welcome! 👍

[@joaofrancisco8864]

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

[@mathemaniac]

Thank you so much!

[@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 😄

[@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.

[@camilo_murcia]

bro, I really appreciate this video, straight to the point and good explanantion

[@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!

[@johnemory7485]

Love the long form videos, keep them coming!

[@viggosimonsen]

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

[@martian0x80]

This was so beautiful, I think I shed a tear.

[@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!

[@Jazzdrummerperson]

Fantastic video! Thanks! Watched it until the end. Keep up the good work.

[@jens6076]

Yes! Watched to the end, highly enjoyable.

[@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 :)

[@Stelios.Posantzis]

Great intro and quick overview of complex analysis!

[@russiainvadesukraine8330]

loved it,keep up the good work, it was like an amazing piece of art, just like a song, even better at 2x

[@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!

[@lourencoentrudo]

You've outdone yourself. Superbly presented!

[@mathemaniac]

Thanks so much!

[@micha2325]

I was waiting for this video! Wonderful explenation of complex integrals, thank you so much :)

[@mathemaniac]

Thanks so much for the support!

[@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.

[@merich4940]

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

[@ksbalaji1287]

Thank you for this beautiful video! It is full of insight!

[@PrismShahriar]

So much useful! Thanks for making this video.

[@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!

[@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 :)

[@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!

[@sagarjc5423]

Thanks for explaining so well. I hope there are many more videos on complex integration

[@gauthierruberti8065]

Thank you sooooo much for this video. This answers pretty much every doubt I had about the subject

[@General12th]

This is a great video with a great ending. I watched all the way through!

[@mathemaniac]

Glad you enjoyed it!

[@filomenamendes5349]

Watched all the way till the end. Very helpful. Well done.

[@vitalysarmaev]

It's just a brilliant explanation! Bravo 👏

[@kev2582]

Watched till the end and probably will watch some parts again Amazing content!!!

[@jameskoh3463]

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

[@brendawilliams8062]

I’m barely passing this one. P and a looks likely the genius with the genius.

[@brendawilliams8062]

Sorry, p and q

[@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!

[@Jonathan1000sKoT]

Amazing video, love the connection of complex integration with the Pólya vector field!

[@CPisco]

What a video. I was struggling with complex integrals but you made it easier. Thanks a lot !!

[@mathemaniac]

Glad it helps!

[@YuriyKrivosheyev]

Great! Thank you for all time you’ve spent on this.

[@mathemaniac]

Glad you enjoyed it!

[@otonki3150]

Made it to the end, amazing video. Thanks

[@douglasstrother6584]

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

[@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

[@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.

[@adarshkishore6666]

I watched till the end. Beautifully presented with nice visuals! When I studied this I was bored, but I knew that there must be some fascinating visualization behind them. Thank you for bringing that out!

[@mathemaniac]

Thanks for the compliment!

[@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!

[@arttrust7629]

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

[@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

[@HaramGuys]

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

[@benniasoff90]

Amazing !!!! Thanks for sharing !!!!