Complex Integrals and Cauchy's Integral Theorem.

This video covers the method of complex integration and proves Cauchy's Theorem when the complex function has a continuous derivative.

Some background knowledge of line integrals in vector calculus is useful to understand connections, but not necessary.

Any questions? Let me know in the comments!

[spasbanchev]

Just wanted to say a quick thank you for all the help throughout various topics around multiple subjects. What I most appreciate about your videos is not only the straightforward, as easy to grasp as possible, approach but also the amount of effort you put in editing your videos to exclude any distractive noises and unnecessary pauses etc. You are doing an admirable job and I can actually see how future teaching could move online with videos like this one allowing me to understand something that my lecturer takes about 1.5 hours to teach. It actually feels like you are saving my time.

[dickdastardly]

These videos have no fat whatsoever. Streamlined, hard hitting stuff. That was Cauchy’s integral theorem in 8 minutes. Earned an instant subscribe from me. Excellent work.

[psychedelicfungi]

Great video. Very clearly explained without labouring too much on the background material. Wish I didn't get so distracted by "Cauchy" being mispronounced so much though!

[luminescence]

Thank you so REALLY HELPFUL

I LOVE THE WHOLE COMPLEX VARIABLE PLAYLIST🥰

[TheLocalSexOffender]

You are an absolute legend, these videos are so useful in making sure my understanding is sound. You have very efficient videos and they are great before an exam. You are a life saver!

[pokpikchan]

Thank you very much
The only lecture that highlighted the green's theorem

[MrTheJevil]

You really butchered Cauchy's name lmao

[typo]

So a contour integral is really just a line integral?

[ninaraspberry]

Truly captivating work, have no idea how you smarties do this stuff but I like to watch

[ZelForShort]

Holy shit that's a alot of integrals. Great video btw. Barely understood anything due to the fact I'm only in precalc but I love the format. Its like a cut commentary for math. LOVE it <3

[shaheedperez]

I just wanted to ask what do you mean by saying that C has to have a finite number of corners because in the case of a circle, which is similar to a polygon with infinite corners, doesn't that say you can't have smooth curves; any help would be much appreciated.

[approachableGoals]

Really appreciate your videos, simple and elegent explanations!

[jaralara]

What do you write on? Your handwriting’s incredibly neat and clear just like the information you present. Subbed

[trejohnson]

wow the illumination of E&M given to me by complex analysis is grand

[margaretbelanger]

Hey, awesome video, great explanations, very helpful! Thank you! If I could make a suggestion, I found that the video moved a little bit fast for me and i had to pause a bit, but other than that awesome job thank you so much!!

[sauravmalik]

I think the curve should be piece wise smoth

[Lezrec]

Is it a valid proof to say that if we assume that f(z) is some derivative dg/dz then if we integrate f(z) from z to z (in some circular path), we essentially get g(z) - g(z) = 0? Essentially the FTC for Line Integrals? I know that infinite differentiability is something holomorphic functions have, but I don't know if that goes the other way...

[dominicellis]

Could you do a video on the complex formulation of the inverse laplace transform. I understand that if gamma = 0, we uncover the inverse Fourier transform, but I don’t understand how you would compute the indefinite contour integral if gamma is non zero. I love your vids btw.

[shreya_sinha.]

i have a doubt ..does cauchys theorem imply that area of contour is zero ? how can that be?

[jhonnatangomes]

I was thinking the following: you can claim that df/dz is continuous simply because f is differentiable inside C as a hypothesis. Since it is differentiable it will also be continuous. The real assumption one has to make here is the one that the partial derivatives of u and v are continuous. Correct me if I'm wrong.