How to find the Residues of a Complex Function

Рет қаралды 199,007

Күн бұрын

In this video, I describe 3 techniques behind finding residues of a complex function: 1) Using the Laurent series, 2) A residue-finding approach for simple poles, and 3) A residue-finding approach for non-simple poles.
I also prove/verify these techniques, which are ultimately going to be used to calculate complex integrals (and even real integrals) when applying the Residue Theorem.
Questions/suggestions? Let me know in the comments! Also, yes, I spelled 'technique' wrong at 8:50. Pls forgive my transgression.
Prereqs: The playlist so far (the first 7 videos, especially the Laurent series and residue theorem one): • Complex Variables and ...
Lecture Notes: drive.google.com/file/d/0B_ur...
Support my Patreon: www.patreon.com/user?u=4354534

Пікірлер: 150

@mahnoorpirzada6638 4 жыл бұрын

i find it incredible that i pay a lot of money to attend a top ranked uni and be taught this, but end up learning all of it for free on youtube. this is amazing, thank you.

@covingtonkua9404 4 жыл бұрын

can't relate more

@FighterAceee94 3 жыл бұрын

And here I thought the reason I wasn't taught this at uni was because my uni is free (paid by taxpayers)

@mettataurr 3 жыл бұрын

More teachers need to learn from this neon-focused study

@SouravBagchigoogleplus 3 жыл бұрын

I also attend a top ranked University. That's why our professor assumed that we learnt these simple things in high school.

@technoguyx 3 жыл бұрын

You don't pay to learn -- you're expected to learn yourself. The point of university is to meet people and find opportunities. And it should be free for anyone capable of entering it.

@tariqandrea398 Жыл бұрын

This is not just a lecture on complex variables and residues. This is a service to the human race. Years back when I took complex variables at Stanford, it was near impossible to understand Cauchy's theorem, residues, and their uses. It took me an extremely long time to understand what this lecture communicated in 15 minutes. Current students from the University fo Toronto, which prides itself as one of the worlds top 20 post some horror stories about their freshmen courses in calculus, physics and YES even pedestrian chemistry and biology.

@rachanasoni007 6 жыл бұрын

Oh my god. This is great. Why I haven't discovered you earlier 👍I just loved your presentation style speaking style. You have earned a fan 🙏

@rikudou766 2 ай бұрын

Thank you Dr. Khan. My proffessor finished up complex analysis within 1.5 month and we tought it was impressive. But with you I understood the course within a night. Please continue teaching us because you are better than any proffessor I know.

@coolbowties394 4 жыл бұрын

Thank you SO much. I have an exam on complex function theory in a few days and my mind had gone totally blank! Amazing, clear, thorough presentation. Thanks for saving my degree!

@azmath2059 6 жыл бұрын

Brilliant, thank you for posting. It's like being in a dark room and having the light switched on.

@georgedelgado1995 5 жыл бұрын

alex zorba 🕺🏻

@charlierandallmoll1531 3 ай бұрын

After hours of confusion with my terrible textbook (fischer), you have cleared everything up for me in less than 15 mins, thank you

@alvinpan7004 3 жыл бұрын

WOW, I'm so astonished by the fact that the contents and explanations are laconic, comprehensive and engaging, excellent in all aspects! Thank you so much for this marvellous course!

@alexrosellverges8345 5 жыл бұрын

This series is pretty cool. Concise, formal and clear, thanks!!

@maithaap4565 4 жыл бұрын

Wow! I have watched this video before. At first, I haven't understood it, but then I watched the whole series (with skipping the proofs) and it is very clear. Thank you sir, your explanation is to the point, and very clear!

@StanDoesaThing 4 жыл бұрын

This video is exactly what I was telling my friend I wanted from my professor but wasn't getting. Thanks!

@sayy_gaarr 4 жыл бұрын

This video perfectly shows why I love math !! Greatly appreciated. Keep it up.

@frechernono6401 4 жыл бұрын

Thank you really much. The explanation is so well done and easy to understand. Looking forward to learn more from your videos!

@LowMoneyStudy 5 жыл бұрын

Absolutely Clean And Amazing Explanation, Thank you

@athul3870 3 жыл бұрын

You, my sir, are what would generally be known as a LEGEND.

@captain_jack8640 4 жыл бұрын

Just to inform you sir You are doing a great work for learners like us. Thank you

@OmarAhmed-ic4fw 3 жыл бұрын

Great work! I wish you would continue with complex analysis making more videos about the theory and examples.

@bhoopendragupta4782 3 жыл бұрын

I was looking for this video all over the internet, nice explanation👍👍👍

@sandracordoba6090 Жыл бұрын

Hope you are getting the recognition you deserve. Great material. Thank you very much for your effort.

@FacultyofKhan Жыл бұрын

Much appreciated!

@kanikagupta6103 6 жыл бұрын

Concepts so beautifully explained

@GoldenPlana 6 жыл бұрын

Your videos are extremely helpful! Thanks!

@ziqizhang1925 5 жыл бұрын

Thank you so much for this high quality video!!

@Zumerjud Жыл бұрын

Very clear explanation. Thank you!

@Frostbitecgi 5 жыл бұрын

i loved this lecture .... all clear to me now .. thank you soooo much

@ryanmckenna2047 2 жыл бұрын

The massive dusty tome I was reading on this has been hard going but your lectures made it easy and enjoyable to understand

@tpthpt5973 7 жыл бұрын

I would like to see an example of the residue theorem where a pole lies on a boundary (not inside, not outside). Great videos by the way!

@FacultyofKhan 7 жыл бұрын

Interesting, I can try that in a future video for sure, and thank you for the kind words!

@uhbayhue 2 жыл бұрын

4 years late, but I got a solution for you! Suppose a pole z0 lies on the real axis while you consider a contour in the upper half-plane. Then, you approximate z0 as z0+ i*p, in the limit as p approaches 0. That is, z0 = lim (p --> 0) [ z0 + i*p]. The addition of i*p removes the pole from the boundary and pushes it into the upper half plane. You can equivalently let z0 go to z0 - i*p to push it into the lower half plane. So replace every instance of z0 with z0 +/- i*p, simplify, and take the limit at the end. This way, you can tackle poles on boundaries of contours.

@ayasaki.pb_787 3 жыл бұрын

Much thanks for proving technic 3. My book does the same by saying it's easy to identify that without show me the proof. Now have the complete picture of computing residues.

@xerxes4849 2 жыл бұрын

Thx for this explanation. Made a lot of things clear for me!

@sanamazarniya8092 4 жыл бұрын

just wanted to say that you're amazing and thank you :)

@BIGPIMPINREELS 3 жыл бұрын

Really good work! Thank you so much for this

@nestorv7627 7 жыл бұрын

these videos are sooo beautiful and entertaining

@FacultyofKhan 7 жыл бұрын

Thank you so much!

@sabariprasad4239 5 жыл бұрын

how it is entertaining...

@mariomasters1 4 жыл бұрын

Excellent lecture series man!

@HaNguyen-zw3xz 6 жыл бұрын

Thankyou so much for posting this video!!!!

@mihaipuiu6231 2 жыл бұрын

Very nice demonstration.

@carlosfelipebedoyariveros4776 5 жыл бұрын

What a beatiful lecture!

@audwindcosta8500 6 жыл бұрын

Awesome Lecture Thank you so much sir

@notcavendish9754 7 жыл бұрын

Thank you so much for your great vidios! Sometimes they are life saving.

@FacultyofKhan 7 жыл бұрын

No problem! I appreciate the kind words!

@pushpamsingh3870 5 жыл бұрын

It is very helpful. Thanks a lot for it.

@rayus77 7 жыл бұрын

Thanks so much Khan. You just saved me hours trying to revise Complex Calculus. Your videos are so concise and easy to understanding. Keep these videos coming :D !

@FacultyofKhan 7 жыл бұрын

Thank you, and I will!

@sajidrizvi4665 7 жыл бұрын

Thank you so very much for this :)

@jack000pumpkin 7 жыл бұрын

Great channel. Thank you so much!

@FacultyofKhan 7 жыл бұрын

Glad you like it!

@tompurcell1499 5 жыл бұрын

I might be missing something here, and technically you are correct, but surely for a function f(z) having a pole of order n, it is unnecessary to multiply the Laurent Series for f about the pole α by (z - α) to any exponent greater than n. Indeed (as demonstrated in your last example) for such a function f, with Res(f, α) the residue of f, Res(f, α) = (1/(n-1)!) lim(z → α) d^(n-1)/dz^(n-1) [(z - α)^n f(z)] For m > n, all that we achieve is more work in determining the residue, as fun as repeated differentiating may be.

@metallicafan97ariana 3 жыл бұрын

AMAZING. many thanks

@ArathonSG 4 жыл бұрын

Great video thank you so much

@ascle9095 Жыл бұрын

Thank you so much, life saver

@matthewscott336 11 ай бұрын

Really great content

@mohamedmouh3949 Жыл бұрын

i like the speed of explaining

@boyteam10 4 жыл бұрын

Very good lesson

@lidkaluczkiewicz420 Жыл бұрын

Thanks a lot! I will pass my finals hopefully.

@frankreashore 3 жыл бұрын

Wonderfully clear description. Huge thanks. What drawing tool are you using?

@dikshantdulal587 3 жыл бұрын

Love it!

@fabiospinelli4179 Жыл бұрын

this i steh first math video i needed to put at 0.75 because you are god dam fast, BTW: nice explanation

@tedk2152 4 жыл бұрын

You are an angel.

@samuelj5890 5 жыл бұрын

ayyyyy you the G my man

@cboniefbr 7 жыл бұрын

Amazing videos!!

@FacultyofKhan 7 жыл бұрын

Glad you like them!

@user-kt3ke7fj2l 3 жыл бұрын

give thanks to you from korea teacher you made me keep dreaming

@monikamishra8591 6 жыл бұрын

Wow...your video was very helpful...but can you tell me the tool of making this video? I mean can you tell me the software you are using for making this video?

@stayawayfrommrrogers 6 жыл бұрын

5:19 When z^2 - 1 is factored, wouldn't that add a second (z - 1 ) term into the denominator?

@arbimalngiang7349 4 жыл бұрын

I like the lecture it is very

@carnival121 5 жыл бұрын

I think I'm being very dim but I don't get the first example. The taylor series of the sin(z) is the analytic part right? I dont get how dividing through by 1/z^2 makes it the principal part. I'm clearly not getting how to construct the Laurent series. Does anyone know of any pages that go through step by step how to get the laurent series without the taylor series shortcut?

@TeslaArabic 6 жыл бұрын

Thank you

@kaishaikh6241 7 жыл бұрын

Does the next video on complex variables exist yet? If not, please consider this a request :) great video by the way!

@FacultyofKhan 7 жыл бұрын

Doesn't exist yet, but I'm making one. Thank you for the kind words!

@ChocolateMilkCultLeader 4 жыл бұрын

Exam in 2 days. Thank you for this

@FacultyofKhan 4 жыл бұрын

All the best!

@sureshkaruppasamy6257 2 жыл бұрын

Please help us to find poles and residue of 1/sin²z

@azfarahsan 2 жыл бұрын

thank you

@mogustew 3 жыл бұрын

Why is it that in the third technique we do not have to the take the limit as z -> z_0 as we did when z_0 was a simple pole?

@youssefbenhachem993 5 жыл бұрын

Amazing

@joliverkozlowski 4 жыл бұрын

Valeu! Thanks!!

@stuartyeo5354 Ай бұрын

great presenter =D

@uimasterskill 2 жыл бұрын

Your proof works if the set of singularities is discrete topologically, because only then you can draw small disjoint circles around them. Are they always like this? Can the set of singularities have an accumulation point, for example?

@user-xi4vy9iw1v 2 жыл бұрын

GOAT!!!

@ryanchatterjee 6 жыл бұрын

Another important technique: If you have a function of the form f(z)=p(z)/q(z), where q(z) has a zero at z=a but its first derivative is nonzero at z=a, and p(a) is nonzero, the residue at z=a will be equal to p(a)/q'(a).

@arturo3511 Жыл бұрын

at 4.02, could you explain why it stops at j=1 for a simple pole ?

@WA-hq6ls 4 жыл бұрын

allow me, but this is fu*king nice!!

@ddiq47 2 жыл бұрын

ur a legend

@Ella20399 4 жыл бұрын

Cant thank you enough

@arbimalngiang7349 4 жыл бұрын

Sir can u provide video to find resume(sin1/z-1,1) plz sir

@tanujabysani2663 Жыл бұрын

How to solve fourier series and transforms

@faiyazmahir5210 4 жыл бұрын

Can I use rule three for every residue function?

@CarlosRamos-tx4er 4 жыл бұрын

well , if lim (z-z_0)f(z) when z tends to z_0 tends to infinity , z_0 also could be a essential singularity but if there is a finite n such the lim((z-z_0)^n)f(z) is finite, then z_0 is nesesarily a pole

@rjbeatz 4 жыл бұрын

Hello! Can you recommend any books related to this topic? Thank You

@GoogleUser-ee8ro Жыл бұрын

5:27 why is z = i is the only simple pole? how about z=-i, z=1 and z=-1? cosz for those zeros are also analytic and they all appear once in the denominator.

@RavindraKempaiah 6 жыл бұрын

You're the best. I wish you could offer a Coursera or EdX course. Are you a faculty at a University?

@FacultyofKhan 6 жыл бұрын

Nope, not a faculty. I'm a university student.

@marcelasiqueira6658 5 жыл бұрын

Most professors at universities do NOT teach so good.

@sakshigupta4293 6 жыл бұрын

How to find residues of trigonometric functions

@madhvisaiya5164 6 жыл бұрын

I want to make similar videos. How do you make them. What app?? Or tricks?

@joy4mkol 2 жыл бұрын

How about residue of (z-Sin z)/z?

@ItsKickey 2 жыл бұрын

"The proof is very simple and left it as an exercises" LOL I am preparing the midterm and all my roommate are looking at me like I am a fool

@pushpamsingh3870 5 жыл бұрын

Please make a video on branch point, too.

@FacultyofKhan 5 жыл бұрын

Working on it!

@NicolasSchmidMusic 3 жыл бұрын

I don't really see why for a pole of order n we can't just multiply with (z -zo)^n. Souldn't it also give us the residue at zo? Obviously what I say is wrong because it doesn't give the same result but I can't find out why. EDIT: I just got it if we multiply by (z-zo)^n we would actually get bn and not b1

@whoisnp3305 Ай бұрын

How do you know if something has a pole ?. Is there a video that you have on this concept ?

@FacultyofKhan Ай бұрын

I define poles here (I'd suggest watching the videos before this one on the playlist so you have a stronger idea of what's happening): kzfaq.info/get/bejne/ZsB_gM6e3pPFhKM.html

@allenfernando4619 4 жыл бұрын

Hi, how is the cosine of z continuous and holomorphic at i ? Can someone explain.

@MrBorderlands123 4 жыл бұрын

Use the Cauchy-Riemann equations to show that cos(z) is an analytic function.

@madalinam7137 3 жыл бұрын

Hi! Thanks for the nice video! I have a question, for the case of finding the residue for a pole of higher than 1 order, let's take as an example a pole of 2nd order. Wouldn't it give the same result and also be simpler if the residue would be calculated as lim z-> z0 { (z-z0)^2 * F(z)} instead of doing so many derivatives? it means that it would apply to a pole of any order n>=1 as follows: R[z0]= lim z-> z0 { (z-z0)^n * F(z)} with n = the order of the pole. What is the difference? Thanks!

@mbrusyda9437 2 жыл бұрын

No, plugging z0 to the series gives b_n not the residue b_1

@jayjayf9699 3 жыл бұрын

Why are we only finding b1 using technique 3 , what about the other bn values for the residue ?

@mogustew 3 жыл бұрын

Only the coefficient b_1 is called the residue.

@lautaro450 7 жыл бұрын

This is so useful.. Thank you! you really help me. I'm student of physics so.. If you need some help in whatever you want.. just send me a message :).

@FacultyofKhan 7 жыл бұрын

Thank you for the feedback and the offer! I don't need help right now but if I do, I'll be sure to make an announcement!