Introducing Branch Points and Branch Cuts | Complex Variables

The video many of you have requested is finally here! In this lesson, I introduce #BranchPoints and #BranchCuts in the context of multiple-valued functions of #ComplexVariables.

Specifically, I describe the natural log function as an example of a multiple-valued function requiring us to 'cut it up' to form a single-valued function which is more amenable to integration. The process of 'cutting it up' involves the construction of a branch cut, which will come in handy when we get to integrating these functions in the next video.

Questions/requests? Let me know in the comments!

Special thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Damjan
Yenyo Pal
Lisa Bouchard
Patapom
Gabriel Sommer
Eugene Bulkin
Adam Pesl
Yiu Chong
René Gastelumendi

[DewyPeters]

Brilliant timing! I was studying this and struggling on branch points and branch cuts for most of today.... and you happen to release this video at this very moment :O

[captainkielbasa]

Excellent! Thank you for honoring requests. My complex var. professor is ~70 and flies through notes without taking a moment to provide a wider perspective of concepts. Thank you so much for all of your series!

[biz]

This is so good! It made no sense before I watched this video but now I get it!

[WorldOf]

I'm just here to support the grind. Hope you're doing well bro!!

[trisharoy]

You are awesome dude. I was struggling with this

[xyzct]

What's interesting about complex analysis is that when one studies it for fun -- outside of the pressures of school -- it is actually quite simple. (Well, maybe simple isn't the right term, but you know what I mean.)

[soohoonc]

Thank you for the visualization, this makes so much more sense now.

[Woodra_YT]

Thank you for clarifying why the end points alpha and alpha + 2pi were not included as part of the branch cuts. Great video^_^

[shihaowang]

AWESOMEEEE You save my life in the course Asymptotic Analysis.

[lorrainerosello]

I'm studying for an exam in my Math Methods in Physics class and this really helped! (The textbook only really had about a paragraph on this :/) Thank you!!!

[somyakumar]

Great video! Despite being a bit fast paced, it was very lucid

[omanshsharma]

you have a gorgeous handwriting!! thanks for the explanation!!

[motiversity]

You're a grt teacher....thanks for these lectures...God bless!!❤

[soundslikemath]

First, thank you so much these are great! Second, I think multiple valued functions are still functions, except we have subtly changed the codomain to the powerset of what we thought it was.

[jaredprice]

You’re amazing man. Would love to see a video on integrating a multivalued function over the dogbone(dumbbell) contour next.

[ksmg-ys]

Absolutely amazing! anything similar about the complex root function?

[ShouEnLin]

thank you for making this helpful video!

[Physics_mania_supriya]

Hi, nice presentation. I can grasp the idea much easier with the help of this video. By the way, could you please tell me which software you are using to write this black board type learning.

[esisimp]

At 1:26 you say we equate the real and imaginary part of both sides and then equate e^u with r and e^{iv} with e^{i\theta} but these are not the real and imaginary parts. Wouldn't the correct thing to do be use Euler formula first and equate e^{u}cos(v) = r.cos(\theta) and same for the imaginary. This might appear pedantic with both approaches leading to the same result but isn't that but isn't this the correct statement ?

[dwightd]

Your interval for theta should be clopen