Residue Theorem and Proof

[@FacultyofKhan]

Well, looks like I was wrong when I said that this is the last Complex Variables video (that's not based on examples/applications) for the time-being, because I just made a new one on the ML Inequality! Check it out: kzbin.info/www/bejne/qXbch5SVjJaGr9E

[@AtiyabZafar]

Faculty of Khan man, great work. can you also make some examples of contour integrals?

[@FacultyofKhan]

I've done an example in the ML Inequality video, but of course, I can do a video calculating a contour integral instead of just finding its upper bound. I'll put it up later when I get to applications/examples in Complex Variables. Thanks for the suggestion!

[@AtiyabZafar]

Faculty of Khan yeah i saw that Video too. the one that was uploaded yesterday . Very good work. Sometimes it's hard to grasp the concept of lemma and i understood it well. so thanks again

[@pcpilot6991]

I am approaching the "retirement" phase of life, and now trying to mess around with advanced physics and mathematics, for pleasure. But I first need to revisit and strengthen my basics, and your tutorials are of immense help; the delivery is at the optimum pace. Thank you so much!

[@oneinabillion654]

Can't believe that passion is still burning in you. I'm on the other side of the spectrum, a 20year old entering adult life. I salute you!

[@pcpilot6991]

@@oneinabillion654 Wish you a wonderful adult life ahead👍; keep away from rate races and the passion will never jade☺

[@pcpilot6991]

@@oneinabillion654 ...rat races...

[@oneinabillion654]

@@pcpilot6991 That's a deep one. Will keep that in mind down the road :))

[@michaelmensah3666]

L

[@coucou67youpi]

Bravo that's by far the best explanation i found for this theorem on internet! Thank you very much

[@FacultyofKhan]

No problem! Glad you liked it!

[@manojprabhakar9111]

Angry comment: THIS IS THE "THE" BEST VIDEO OF RESIDUE THEOREM IN THE INTERNET😡😡😠😠😡😡

[@martyalden]

I really love this channel, I always come here to review this proof again and again, and I am always amazed by your way of explaining it, love this channel!

[@cosmoshivani]

right!!?

[@Zxv975]

Great lecture. I'm really enjoying going through these concepts, as I really didn't understand them very well in my complex analysis course years ago. These lectures are perfectly succinct, and you've basically been able to cover half a semester's worth of content in an hour, which is incredible.

[@FacultyofKhan]

Thank you so much! I'm going to be adding two more lectures to my complex variables playlist focusing on examples/applications, if you're interested in more (so now I'll be covering even more material in 1 hr 20 minutes). In either case, your kind feedback is much appreciated!

[@marcovillalobos5177]

I really love the rythm in your videos, going straight to the point but without losing anything 10/10

[@m.kamalali]

keep going man u will make difference

[@FacultyofKhan]

I definitely will! Thank you so much!

[@commodorekitty]

Thank you so much for making this series. I had to do a review of complex analysis because I needed to use the Residue Theorem in one of my master's classes in physics. This makes reviewing background material less painful.

[@lr9077]

I came to the comments section looking for angry comments and passive aggressive responses! lol! You're so nice though! Thank you for this beautiful and thorough explanation. I admit that I still don't get it but I'm only now taking general Analysis for the first time. Once I get to complex analysis, I imagine this will make pure and complete sense! Thank you again! Best wishes to you!

[@PunmasterSTP]

Residue theorem and proof? More like “With all these great lectures, I really want to raise the roof!” Thanks again for all of them.

[@rikudou766]

Thank you. I have been studying on this all day from the book and I couldn't for the life of me can't understand mathematical jargon efficently. But you made me understand it with just logic. So thank you sir.

[@scitwi9164]

The visual proof is ingenious! :D

[@richardgivhan8401]

Thank you for making such excellent videos! Would not be able to keep up with my classes without you.

[@Terieni-q7c]

Amazingly clear yet not without rigor!

[@kevinjason6697]

Brilliantly explained! Could do some challenging problems involving these?

[@FacultyofKhan]

Thank you, and absolutely, I'll put 'em on my to-do list! I believe people have requested them before as well for the Residue Theorem, so I'll get on them when I get the chance!

[@aoimiyazaki6291]

Nice explanation! Though you have to keep in mind that this only works for contours of winding number 1 around these singularities.

[@FacultyofKhan]

You're correct. Though I do mention simple closed curve (i.e. winding number 1) in the proof for my Cauchy's Theorem video: kzbin.info/www/bejne/p4WndXmDdtlqoJYm34s And I make use of Cauchy's Theorem when proving the Residue Theorem (at 1:12), but yes, I should have made it more explicit that C was a simple closed curve in this case. Thank you for the feedback! I'll update the description.

[@dijkstra4678]

You may be a little fast but that's what the rewind button is for, thanks for the great explanations as always.

[@yuanyin3383]

So rigorous and clear! Your videos are amazing!!!

[@marlonbrade9004]

at. 5:34 , its not that obvious that it is equal to 2pi i b11 using cauchy integral formula but if you rewrite the integral in terms of theta using the change of variables mention above such that z = z1 +p1 e^itheta , then one can easily solve it.

[@jeanette123321]

How did you get 2*pi*i*b_11 by the use of Cauchys integral? (5.28 in the video)

[@FacultyofKhan]

I got it from Cauchy's Integral Formula, which I cover in another video: kzbin.info/www/bejne/Znu9fqyrqaeeb9E (6:25 in that video is where I write the formula; if you want the procedure, I recommend watching the whole thing)

[@duckymomo7935]

angry and passive aggressive lol

[@eamon_concannon]

5:34 It looks like this part comes from using cauchy's integral formula for the constant function f(z)= z_1 on any closed curve surrounding the point z_1. Since f(z_1) = z_1 we finally get ∮ 1/(z - z_1)dz = 2πi.

[@marlonbrade9004]

yeah and my problem at that certain moment is that z1 is a singular point of f(z).

[@David-zf5lc]

Very gentle explanation. Thank you so much!

[@bannybanny5811]

The Lecture Notes for this video is the notes for the Laurent Series of Complex Functions video.

[@haricashravi3900]

Hello, I have a question : Why didn't you use the derivative formula to show that the principal part of the Laurent series (excluding 1 ofc) goes to zero? It would've been a lot more easier

[@hannalinn640]

Just want to say youre great and I really liked this serie (pun intended)! Thank you!

[@FacultyofKhan]

Haha thank you! Glad you liked it!

[@Mryeo5354]

just started watching your stuff. really enjoying it. thanks =D

[@keerthanab6300]

Thanks for the video sir. Which app are you using to do the derive or write the problem

[@cosmoshivani]

i always wanted to understand the proof of this theorem. thanks to you, now i do know ❤❤🔥

[@tuckerhartland1001]

In the first sum that you display in the video, should the index j run from 1 to n rather than 0 to n? Given that f(z) has n singular points z_1, z_2,...,z_n.

[@FacultyofKhan]

You are correct. I'll make a note in the description. It's fairly minor and hopefully shouldn't kill the video lol. Regardless, thanks for pointing it out!

[@zeinabzeitoun1983]

You're amazing! The best explanation so far!

[@kumailahmad2423]

you are also lovely

[@Tclack]

There might be something I'm missing. I've found that a lot these residue theorem proofs use deformation of contours to show the larger contour equals the sum of the smaller contours around some smaller contour (any size and that makes sense to me) but when evaluating the little contours, it's always assumed it's a circle of radius 1 thus the 2pi term (from the total arclength of a circle. But if you have a cluster of points that are close together then you can't draw a circle that doesn't enclose the other singular points. The only resolution I can think of is that superposition applies. Integrals are linear operators afterall, as are line integrals. I'm not sure if it applies here or if it's a simpler explanation

[@LilithLuz2]

i believe its a circle of constant, but not specifically defined radius, which approaches 0 usually, so its a very very small circle

[@Tclack]

Yeah, you're totally right. I think I figured it out a few days after making the comment, but decided to leave it on because engagement is important for great channels like these. The general substitution should be z=re^iθ dz=ire^iθ dθ

[@akarshchaturvedi2803]

You're good enough to run your own Academy. PS: And yeah, you're awesome!

[@FacultyofKhan]

Thanks man! Appreciate it!

[@Inspireyourself7]

This was really good! Thanks so much.

[@FacultyofKhan]

No problem! Glad you liked it!

[@justanotherasian9711]

Please someone tell me how the 2.pie.i part comes at 5:29

[@zukamot]

¡¡muchas gracias!! eres muy claro para exponer, te agradezco tu esfuerzo, y te deseo muchas felicidades

[@peterrossi6462]

Hi, I'm an electrical engineering student taking an advanced complex variables class. We just learned about the Cauchy Residue Theorem and now we're using this to compute inverse Z transforms. I know it's been a while since Khan academy has added to the complex variables series but is there any way you could do a video explaining how to apply this theorem to the inverse Z transform? It would be incredibly helpful.

[@FirstnameLastname-bx4zk]

How hard is electrical engineering

[@xyzct]

Well done! Love the tempo.

[@valentinahallefors3976]

This is awesome, thanks for making these videos!

[@FacultyofKhan]

No problem! Thank you for the feedback!

[@kctsou821118]

This video really helps a lot!

[@MsMilicaS]

how comewe can't use Cauchy's theorem to say that the second part of Laurent series is zero too

[@fluffiddy6515]

Are you zero because we’re integrating along a closed countour? Or are we integrating along a closed contour because you're zero ?

[@Estoniran]

youre the best!!!! thank you so much for these vids

[@estebanlopez1701]

Hey man, thanks for the video, great job. I was wondering what your major was.

[@karm00n29]

where is the 2pi i from?

[@yvangogh6655]

what if the set of singularities is infinite but discrete (closed with no accumulation points)? would the sum be infinite then?

[@rajbirvirdi4571]

I don't understand why you made the analytic part of the Laurent theorem zero

[@brokodelove3731]

speed of the lecture is very fast....

[@FacultyofKhan]

Thanks for the feedback! I've slowed down the pace a bit in my newer videos. Hopefully it works.

[@laxmanegalitarian8581]

Haha, I did so. In this and previous video.

[@tevinheath3997]

Lol, I listen to it in 1.5 speed 😅

[@NicolasSchmidMusic]

You can pause and ponder a any moment. Personnaly, I watch it at 1.25, and every time I don't get something I pause, maybe go backward, watch again. That's the big advantage of videos in compare to physical lectures, you can do learn everything at your tempo

[@GoogleUser-ee8ro]

I usually speed up the video by 1.5-2x on youtube, but for this video I have to slow it down to 0.75 to follow!

[@lolzomgz1337]

But, wait, can't you make each integral around a singular point be literally any number you like by choosing the radius of the circle you integrate around?

[@FacultyofKhan]

No, because according to Cauchy's Theorem (see: kzbin.info/www/bejne/p4WndXmDdtlqoJY), the contour integral of f(z) around a simple, closed curve C is 0, as long as there are no singular points on or inside C. It doesn't matter how large or small C is; the answer will be zero *every time* as long as there are no singular points on or inside C. In the Residue Theorem, the only contribution to the integral is from the singular point (since the rest of the area gives essentially a zero contribution if you use the Cauchy Theorem logic), so it doesn't matter how big your radius is; your integral around the singular point will still be the same. Hope that helps!

[@jameyatesmauriat6116]

Why you prepare this beforehand because it’s quick explanation and I can’t follow along,please slow down ..

[@ldb579932]

Great presentation but I think there's a minor error in the first 30 sec of the video. You wrote the singularities z1,z2,...but indexed the sum starting from j=0.

[@paulswillkim3744]

Hi , first of all thanks for helping me a lot to get a better interpretation . I just watched your video series about complex function and It really help my understanding in the complex analysis. While i was studying for Expansion of complex function, I got a feeling that it is quite similar concept of linear algebra decomposition. Since it is a linear combination of null space and column space? that is similar to analytical parts and singular parts? So if i understood correctly, then finding residue is the problem of finding eigen values?? (I am from math finance )

[@FacultyofKhan]

I'm not quite sure how finding eigenvalues and finding residues are similar. They seem pretty separate concepts to me. Perhaps I'm missing something?

[@paulswillkim3744]

sorry it was a silly question.. after reviewed a few times. I realised that they are different concept... and do you have a plan to solve a inverse fourier transform using residue..?

[@frankiedore5263]

As a person who just stumbled upon this what exactly is this math used for?...what is the real world use and application of this madness?

[@FacultyofKhan]

The Residue Theorem is typically used to compute difficult integrals that conventional techniques would make very difficult to determine. For example, see this video: kzbin.info/www/bejne/l2PSlKWva7F1m5I

[@bathiniakash6046]

great video sir!!, I wish you taught me in college.

[@OmarAhmed-ic4fw]

Great explanation!

[@FacultyofKhan]

Glad it was helpful!

[@pratyushkumar7019]

Really nice sir

[@icant_thinkbetter]

a fan of this guy :D thanks man

[@zubairbeig2702]

Good information 👍

[@zeluizsn869]

Very clear!

[@soilammirza9198]

Thanks so much sir

[@laxmanegalitarian8581]

I found your series the best video series explaining complex variable problems. I however have a complaint. You do a whole lot of things in a 7 min video. That is not fair. You make me pause the video uncountable number of times in this little 7 min time.

[@aarifhussain3700]

Nice sir u r mine favorite arif Hussain from Pakistan

[@vsmk8747]

don't use 'j' if 'i' is also being used.as at times in electrical problems j is used to denote root-1 in place of i

[@FacultyofKhan]

Yeah but I think you could argue that a lot of letters get repeated for the same mathematical quantities in different disciplines. 'j' is used as an index when 'i' isn't available (which is what I did here), and frankly, I (hehe) think that 'i' is a better letter to represent the square root of -1 (i for iota). You do have a point though; perhaps I could mention this distinction in a future video. Thanks for the feedback!

[@stepankovacic2658]

Very nice

[@TheSaralIAS]

include example man !!!

[@pettPette]

good

[@FacultyofKhan]

Glad you liked it!

[@asmarali8579]

great sir

[@frankiedore5263]

Sorry..this is way beyond me. I find its complexity alone mind blowing but referring me to another complex video to explain the one I asked about didn't help..lol...i just wanted to know how its used in the real world?..like what does it get used for?...(example answer [" well its used for calculating building failure figures during earthquakes" or " its used for the programming of giant automated robotic assembly lines"...that's how far I was hoping you would dumb it down for me...pls😀

[@FacultyofKhan]

In that case, it's used in Quantum Mechanics (and other areas of Math/Physics) to calculate definite integrals. I'm not aware of any industrial applications where it comes up though.

[@bluemonster1354]

digital signal processing ; to calculate the inverse Z transform

[@rachelwilson-farrer8787]

Very good at explaining but goes soo fast 😬

[@willyou2199]

[@Studio-gs7ye]

You uploaded wrong Lecture Notes.

[@FacultyofKhan]

If you go on the right side of the image and down, you should see the notes for this video (Residue Theorem). Apologies for the confusion; I put notes from two lectures into one image!

[@amaljeevk3950]

❤

[@theambassadorstutors]

thumb up

[@davidenatale5956]

Gg thanks

[@photon2724]

you lost him at 1:35

[@bleezy6]

Insert angry comment