Complex Analysis 13 | Complex Logarithm

[@brightsideofmaths]

Please substitute "stripe" with "strip" every time I say it :)

[@andrewtaylor9799]

Great video. Thanks for the geometric intuition; I needed a refresher. The complex exponential is often very convenient for modeling radio frequency communication signals in electrical engineering applications.

[@trogdorbu]

Love this! One small thing to help disambiguate your explanation in the latter half of the video: use the word "strip" instead of "stripe". "Stripe" connotes an element of design/style and without consideration of size, whereas "strip" connotes a generic element almost always of a particular width. Compare the term "critical strip" in the Riemann hypothesis. So in both the right and left hand planes, you've used green "stripes" (design element, width irrelevant) to style the areas of interest, but on the left hand planes, it is the "strip" (section of fixed width) that falls between iπ and -iπ that is what is mapped to by the logarithm.

[@brightsideofmaths]

Oh, thank you very much. I was not aware of this difference :)

[@seankim9494]

I just wanna say thank you for your channel, I've been watching your manifolds series and it's an absolute masterpiece.

[@brightsideofmaths]

Thank you very much!

[@mfarae2504]

This video series is absolutely amazing. I am taking Complex analysis this semester as an engineering major and your videos cover everything we studied so far. If possible, please do make videos on next topics such as complex integration like Cauchy integrals, recap on path integrals, etc.

[@brightsideofmaths]

They will definitely come :)

[@nikosspanos1036]

WTF man! You explain things that have never been mentioned to my university. I love you !

[@alexanderkononov1862]

Fantastic videos both in English and German, defintitely will watch a lot of your courses & videos in the future. Keep it up!

[@brightsideofmaths]

Awesome, thank you!

[@gutzimmumdo4910]

cool video altou from 7:19 i think theres a bunch of asumptions like; "what is a branch", "why does that vertical line gets map to the circle" maybe i missed or is goin to get covered latter dono, thanks anyway great course so far.

[@brightsideofmaths]

We know the complex exponential function from former videos very well :)

[@AJ-et3vf]

Great video! Thank you!

[@ZeothGames]

"Ah yes, such a simple function. Let's make every question in the exam so that you need to use this function very creativly for a chance to solve it!" My Professor probably.

[@brightsideofmaths]

That was not me :D

[@arghyachakraborty1151]

Amazing explanation!It would be really helpful if you recommend some interesting book...

[@brightsideofmaths]

In my opinion every book about complex analysis is good. Just pick the one that suits your style.

[@darthtleilaxu4021]

Merci !

[@OvercookedOmlette]

Impeccable video! I wanted to ask why having the pink lines at iπ and -iπ translates to pink line on the negative side of the real numbers?

[@brightsideofmaths]

Thanks! It's the definition of the complex exponential function.

[@olivermechling7975]

10/10 explanation

[@brightsideofmaths]

Very nice. Thanks :)

[@AHeroOfAzeroth]

Hi thanks for the video! I have a question I understand e^iy lies on the circle but what about e^xe^iy this doesnt lies on the circle right?

[@ridnap]

I love your videos! Thank you very much for your content, it helps more than you can Imagine! I dont know if the comments of the Complex Analysis series is the right place to tell you this, but I have really been missing a video about the Riesz Interpolation Theorem for example in the fu ctional analysis series. Do you plan on adding further episodes to older series or are they finished?

[@brightsideofmaths]

Thanks! I still add new videos to functional analysis :)

[@Jaylooker]

The complex exponential map looks like a circle inversion

[@StratosFair]

Why not include, say, the line y=iπ in the domain of exp ? That way it's image is the whole complex plane (minus zero) and the function is still bijective. Is it because we absolutely want the domain to be an open set ?

[@StratosFair]

Ah, nevermind, just saw the end of the video about the continuity of log

[@MathCuriousity]

hey just learning about "complex" stuff and I have a question: When we are given a question with say a complex root we need to find or a complex log, which one of them is actually defining the other where when we have that one as a question, we must choose the inverse's tree branch cutting?

[@brightsideofmaths]

Complex root means a given branch usually.

[@MathCuriousity]

@@brightsideofmaths right but I watched your logarithm video (and admittedly I haven’t had any serious complex exposure), and I am just wondering: is the complex logarithm defined based on defining the complex exponential or is the complex exponential defined based on the complex logarithm?

[@brightsideofmaths]

@@MathCuriousity We first defined the complex exponential :)

[@MathCuriousity]

@@brightsideofmaths Ah ok! It probably is a bad idea for me to jump right to complex logs and complex exponentials before learning complex analysis from beginning right? I honestly was upset that I could not take the log of a neg number when learning about logs and I went down the rabbit hole and it led me here! I am going to continue my quest to understand complex log ! So neither the complex exponential nor the complex log are well defined right? So the whole point is finding a way to make them well defined? Or is it that we want to make complex log and complex exponential inverses of one another? (Or is it both)? Thank you Love you!

[@brightsideofmaths]

@@MathCuriousity Of course, we define the functions in the correct way here. Just watch the first 14 parts of my Complex Analysis course :D

[@benbadman123]

Im slightly confused on the way you explained the periodicity of the exponential for complex values. What did you mean by we need to 'define' PI/2 such that it is the smallest real x such that cos(x) is 0? Isn't this already the case?

[@brightsideofmaths]

We define the number Pi in this way :)

[@benbadman123]

@@brightsideofmaths Ahh haha true. Thanks!

[@sarkarsubhadipofficial]

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