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 RUclips decides that a similar comment is spam lol] Please refer to the previous video for why I skipped the differentiation part: ruclips.net/video/F-kYuvSyC-A/видео.html 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: ruclips.net/channel/UCWdGv5veEBYCn99pT7XJsjwvideos
Oh my God dude... I have been literally waiting for YEARS for a RUclipsr to make a professional visualization of this!! ... Words can't express my thanks... 🙏
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.
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.
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.
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 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.
@@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.
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?
@@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. :)
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!
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! ❤️
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!
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!
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
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.
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!
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...😌😌😌
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
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.
Such an amazing video, possibly one of the best vid of complex integration outhere, great job! Btw, is this inspirated on the book Visual Complex Analysys from Tristan?
Ohh, I only saw this video, I'll check the rest of the series when I have time. Hope one day I can animate my videos, as well as you, did in this one, 3B1B vibes here hehe, cya mate!😁
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!
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 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!
there is an error in chapter 2 at the time stamp 9:38 .. Although you're adding up the angles between f(z) bar and dz which corresponds to the value theta but the animation on the first line didn't change alpha+beta to theta instead it was shown multiplied
Que buen video, felicidades en mi caso lo relaciono con las leyes de Maxwell ojalá así me lo hubieran enseñado en mi clase de cálculo de la universidad, felicidades de nuevo.
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 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.
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.
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!
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.
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
Thank You for this really beautiful, very didactic and enjoyable video on Polya Vector Fields, Complex Analysis, Cauchy Riemman, Complex Integration, Vector Analysis. It would be great if You can create a Part II version continuation of this specific topic using Matlab + Simulink , Computing Symbolic Math and other CAD tools with applications. [Engineering, Physics, Quantum Physics, Chemistry, etc...].
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!
I study physics and this play list helped to understand my semester of complex analysis, it was really nice/cool to understand better what those excercises were about 😂
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 🙏
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 💯
Well, I had my final complex analysis test last Monday, so watching this video right now is a nonsense for me hahahah. However, it's an amazing video. I never understood why the residue is a_(-1) until now. Thank you!!
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 😄
I've been trying to work on complex integration for a couple days now, and I got stuck so I checked if I could find a video somewhere (I didn't remember 3B1B making any in depth video about this topic so I thought I'd look elsewhere). Seriously, this is explained really clearly and I couldn't have hoped for better contebt than this, it's reall been motivating me to work more on all this. I've been going through your other videos and there are gems that are definitely worth subscribing for! (the geometric interpretation of sin(x)'s Taylor series really surprised me, I've been working with it for years and never would have seen that coming)
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.
Hi! I’ve got exam of complex analysis the day after tomorrow. Your video really helped to clear things up, thanks! But I don’t like the cosine example because this is one where the infinite arc part doesn’t disappear and you need the extra trick by using de moivre…
Glad it helps! But the arc part does disappear! The estimation lemma already works to say that the integral on the arc goes like 1/R, which goes to 0. If you have learnt it, Jordan's lemma would make this part a lot easier!
@Mathemaniac What I meant is that you have to be smart and change the cosine into the exponential function because trying to integrate the cosine directly doesn’t work. Or does it? When I tried it naively I got e^-1 +e in stead of the correct 1/e
I thought Polya vector fields looked fishy, but seeing the dot product added to an imaginary cross product revealed why. Nowadays, when I see a dot and cross product together, I immediately think of the geometric product. Treating the complex numbers like GA complex numbers in G(2), converting to a Polya vector field seems equivalent to converting from a complex number to a normal vector by right-multiplying by the x-axis unit vector. To be clear, the geometric product between two GA complex numbers (or GA quaternions in G(3)) is identical to normal complex multiplication, while the geometric product between two vectors gives a complex number (in G(2)). So... ∫f(z)dz = ∫∇FdA = Work + iFlux
2 месяца назад
Absolutely amazing video. Any following planed? For instance, on analytic continuation : )
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
Very impressive video! For the discussion in Chapter 6, may I say that, only charge matters, and those integrals around dipoles or quadrupoles and so on will vanish. That's quite interesting and is there any related physics in this?
Nice video ! ‘though it's important to point out (at 8:00) that the cross product results a vector (not a scalar). ∇ × ( iu − jv ) = −k(∂v/∂x + ∂u/∂y) if ∇ × ( iu − jv ) = 0 then ∂v/∂x = −∂u/∂y
I have a doubt... I want to find areas of regions between certain points in a complex plane (imagine a grid) that is transformed. Because I want to compare euclidian areas to transformed areas by a certain holomorphic function. How can I do it? I mean if a function is holomorphic the area will be cero because of Cauchy's theorem, right? Then, how I calculate this areas. I am thinking about functions like e^z. Someone has any ideas?
"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!" Hmm... I'm not a fan of this. Either reveal it right there where you were talking about it or don't talk about it at all. Don't dangle it like that. I will currently assume you are using 3b1b's engine since it looks similar. Great coverage of this topic btw.
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.
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
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[hopefully, this time, the pinned comment won't disappear like last video, cos for some reason RUclips decides that a similar comment is spam lol]
Please refer to the previous video for why I skipped the differentiation part: ruclips.net/video/F-kYuvSyC-A/видео.html
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: ruclips.net/channel/UCWdGv5veEBYCn99pT7XJsjwvideos
Oh my God dude...
I have been literally waiting for YEARS for a RUclipsr to make a professional visualization of this!! ...
Words can't express my thanks...
🙏
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.
Very impressed with this complex analysis series, well done!
Thank you so much :)
I agree, it was so clear, beautiful and insightful
3B1B quality
Hi trefor
Your videos are incredible as well!
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.
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.
@@RAyLV17 Ah the crossover between mechanical and electrical :-)
@@nosnibor800 now we have internet we can make our knowledge more deeper. Easy life 😌
I think used in Nyquist Criterion
Im a freshman in college for Electrical Engineering, looking forward to the challenge of figuring this stuff out.
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!
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.
@@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
@@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.
Flashbacks dude, flashbacks. But on the other hand, there were worse things in control theory classes.
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.
Glad you enjoyed it!
@@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.
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?
@@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. :)
@TheTKPizza Ohh so the antiderivative is basically how much distance was covered? Or how much distance the object has had from its origin?
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!
By far my favorite video covering complex analysis, ever. You put together nearly all the integration topics so well!
Glad to hear it!
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! ❤️
i watched to the end, i learned a lot of what i wanted to learn in this episode!!
Glad you like it!
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!
Thanks for providing the details here!
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!
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
Yes indeed! Though obviously in "the real world", we wouldn't see this connection very well, because we live in 3D!
@@mathemaniac Did you imagined everything before making this video or you visualised everything using software first .......?
This single video feels like a crash course on Complex Analysis
Yes, indeed that's the intention.
Ur comment piqued my interest
I love everything related to Cauchy's mathematical topics. Now I love this Channel.
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.
I love that the ads are between segments instead of annoying me while he's explaining
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!
Glad it helps!
why wait 1.5 years when you can start now!
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...😌😌😌
Thanks for the kind words!
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
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.
Such an amazing video, possibly one of the best vid of complex integration outhere, great job!
Btw, is this inspirated on the book Visual Complex Analysys from Tristan?
Thanks for the compliment! I did say in the beginning of the video series that this entire series is inspired by that book.
Ohh, I only saw this video, I'll check the rest of the series when I have time.
Hope one day I can animate my videos, as well as you, did in this one, 3B1B vibes here hehe, cya mate!😁
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!
Glad you like them!
@@mathemaniac :>
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!
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.
@@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!
The hardest thing about complex analysis is proving the Jordan Curve Theorem
Thanks. I am electrical engineer studied engineering 4 decades back. This video seamlessly put all the maths concepts together.
movies like this one should be shown to school-children, beats any textbook by far
there is an error in chapter 2 at the time stamp 9:38 .. Although you're adding up the angles between f(z) bar and dz which corresponds to the value theta but the animation on the first line didn't change alpha+beta to theta instead it was shown multiplied
Literally took this class last semester, WHERE WAS THIS VIDEO THEN
Watched the whole thing, it helped me enourmously the week before my complex analysis final!
I learned another method to find residues first that I find more efficient. Taking the limit as z -> a of (z - a)f(z)
Que buen video, felicidades en mi caso lo relaciono con las leyes de Maxwell ojalá así me lo hubieran enseñado en mi clase de cálculo de la universidad, felicidades de nuevo.
This the one of the if not the best complex analysis explanations I've ever seen !, thank you for your efforts .
Glad it was helpful!
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.
Thanks so much for the appreciation!
@@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.
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.
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.
bro, I really appreciate this video, straight to the point and good explanantion
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!
Really glad that you like the video!
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.
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
Amazing! Every second of this video was well worth it, will surely rewatch it again to internalize all those ideas. Thank you!
Preserve your sanity! We are watching (to the end)!
Very good video :)
Thank You for this really beautiful, very didactic and enjoyable video on Polya Vector Fields, Complex Analysis, Cauchy Riemman, Complex Integration, Vector Analysis. It would be great if You can create a Part II version continuation of this specific topic using Matlab + Simulink , Computing Symbolic Math and other CAD tools with applications. [Engineering, Physics, Quantum Physics, Chemistry, etc...].
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!
40 mins??? the gods have blessed us with more amazing content bois!!
Best thesis explanation of calc stated in the first minute I've ever heard.
whew, new mathologer video and new mathemaniac video in the same day. excellent
Haha it just so happened that we both upload on the same day.
I study physics and this play list helped to understand my semester of complex analysis, it was really nice/cool to understand better what those excercises were about 😂
Great to hear!
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 🙏
Hope that you will enjoy it!
Wow, this is a great intro to complex analysis
I'm still here, pleas do more videos like this one!
This is absolutely phenomenal, thank you so much for your work!
I’ve watched until the end. Really great content… How have I not subscribed before?!
Thanks for watching till the end!
You've outdone yourself. Superbly presented!
Thanks so much!
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 💯
The person speaking makes this 150% more intresting & understandable
Well, I had my final complex analysis test last Monday, so watching this video right now is a nonsense for me hahahah. However, it's an amazing video. I never understood why the residue is a_(-1) until now. Thank you!!
I love every bit of this video! Absolutely amazing!
Hey, I watched it till the end! When it was over, I hadn't even realized 40 minutes had passed. I wish you had done the last caculations!
Thanks so much for staying till the end!
Got to 40:00
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 😄
Thank you for the video, it helped me so much :)
Glad it helped!
@@mathemaniac are you going to do the continuation of this one ?
I watched till the end :) This video made me even more excited for when I’ll take complex analysis in college, really enjoyable!
Thanks so much! Not many people will watch till the end!
Keep making videos like this
Thanks for the kind words!
Nice video, i watch to the end
Really thank you 🙏, from India.
I completed my M. Sc in mathematics. Your contents are helping me.
Thanks for the video! amazing content and insight in to complex integration!
Glad you liked it!
I have watched the whole video :)
I've been trying to work on complex integration for a couple days now, and I got stuck so I checked if I could find a video somewhere (I didn't remember 3B1B making any in depth video about this topic so I thought I'd look elsewhere).
Seriously, this is explained really clearly and I couldn't have hoped for better contebt than this, it's reall been motivating me to work more on all this.
I've been going through your other videos and there are gems that are definitely worth subscribing for! (the geometric interpretation of sin(x)'s Taylor series really surprised me, I've been working with it for years and never would have seen that coming)
Hello there I watched to the end! It was the most intresting! I look forward to more videos like this
Thanks!
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.
Bro love your videos. Keep it up 👍🏼 I stayed until the end.
Thanks for sticking around till the end :)
Hi! I’ve got exam of complex analysis the day after tomorrow. Your video really helped to clear things up, thanks! But I don’t like the cosine example because this is one where the infinite arc part doesn’t disappear and you need the extra trick by using de moivre…
Glad it helps! But the arc part does disappear! The estimation lemma already works to say that the integral on the arc goes like 1/R, which goes to 0. If you have learnt it, Jordan's lemma would make this part a lot easier!
@Mathemaniac What I meant is that you have to be smart and change the cosine into the exponential function because trying to integrate the cosine directly doesn’t work. Or does it? When I tried it naively I got e^-1 +e in stead of the correct 1/e
You write iota× flux there but iota is already contained in flux then how
The flux should be just the integral of the dot product - the vector involving i just means the direction of that complex number.
Pi/e
Yours are some very rare and precious video, i admire you.
Thanks so much!
This is a great video with a great ending. I watched all the way through!
Glad you enjoyed it!
Exactly what I needed in order to fully understand the concept... It's amazing !!
wow i had an exam on this last month, this could've helped me get a few extra points lol good vid
Thanks for explaining so well. I hope there are many more videos on complex integration
I se the video útil the end
comment for sanity ^^
Watched the whole thing! awesome video on complex analysis
I am soooo impressed with uour teaching I super like it.
I'm glad you like it
Yes i watched it whole
Really nice 👌
ghostrunnindeaththrowshadows 🥃🥃🐝🐝 chile in the desert panama//storm 🎹🎹🎹 london40
I love when we make sure that we get the small assumptions right. Thank you, and Bravo!
watched all the way to the end so you can keep your sanity! ;)
Thanks!
thank you so much
Really cool video :) Never saw these things explained in terms of the Pólya vector field before
That's why I made this video! The Pólya vector field makes this way more visual!
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
Thank you so much!!
I thought Polya vector fields looked fishy, but seeing the dot product added to an imaginary cross product revealed why. Nowadays, when I see a dot and cross product together, I immediately think of the geometric product. Treating the complex numbers like GA complex numbers in G(2), converting to a Polya vector field seems equivalent to converting from a complex number to a normal vector by right-multiplying by the x-axis unit vector. To be clear, the geometric product between two GA complex numbers (or GA quaternions in G(3)) is identical to normal complex multiplication, while the geometric product between two vectors gives a complex number (in G(2)). So... ∫f(z)dz = ∫∇FdA = Work + iFlux
Absolutely amazing video. Any following planed? For instance, on analytic continuation : )
wow!!! ur video is ******* beautiful and intuitive!!!!
Glad you liked it
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
Very impressive video! For the discussion in Chapter 6, may I say that, only charge matters, and those integrals around dipoles or quadrupoles and so on will vanish. That's quite interesting and is there any related physics in this?
Nice video !
‘though it's important to point out (at 8:00) that the cross product results a vector (not a scalar).
∇ × ( iu − jv ) = −k(∂v/∂x + ∂u/∂y)
if ∇ × ( iu − jv ) = 0 then ∂v/∂x = −∂u/∂y
I'm still here and enjoying the excellent video
Thanks for the support!
I have a doubt... I want to find areas of regions between certain points in a complex plane (imagine a grid) that is transformed. Because I want to compare euclidian areas to transformed areas by a certain holomorphic function. How can I do it? I mean if a function is holomorphic the area will be cero because of Cauchy's theorem, right? Then, how I calculate this areas. I am thinking about functions like e^z. Someone has any ideas?
"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!"
Hmm... I'm not a fan of this. Either reveal it right there where you were talking about it or don't talk about it at all. Don't dangle it like that.
I will currently assume you are using 3b1b's engine since it looks similar.
Great coverage of this topic btw.
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.