Why imaginary numbers are needed to understand the radius of convergence

[@zachstar]

New STEMerch Store: stemerch.com/

[@masternobody1896]

stupid people: ? genius people: { }

[@uttamkumbhat7991]

Please make a video on what is mechatronics and its future

[@mr.knight8967]

MATH QUESTION Numerical / algebra qué. kzbin.info/www/bejne/qaCxZoGjoMaLm68 One time see

[@darshan7267]

im confused, for stability sigma shud be negative decaying , but region of convergence says it shud be positive . whats happening?

[@sleepycritical6950]

I gotta ask, why does it stop any other values outside of the radius like any value not +/-i? For example the Dirichlet series has an abscissca of convergence instead of a radius where it converges for all values past the singularities, that makes sense. Why is the Taylor series however, a radius?

[@AndrewDotsonvideos]

Imaginary numbers? Jeez when are we ever gunna use this stuff!?

[@phillipgrunkin8050]

wHaT dO yOu MeAn?¿?¿? iMaGiNaRy NuMbErS aRe InCrEdIbLy UsEfUl FoR tHiNgS LiKe SiGnAl PrOcEsSiNg!¡!¡! (Btw love your channel and the math community in general, you guys are so tight-knit)

[@lilithshopping7904]

i and j are your best friends in college....

[@AndrewDotsonvideos]

@@phillipgrunkin8050 :)

[@robertmorrison1657]

@@AndrewDotsonvideos Nice to see you on one of Zachs videos.

[@thephysicistcuber175]

Imagine not being able to Wick rotate.

[@xenontesla122]

That also explains why it's called a "radius" of convergence instead of just a region of convergence. Cool!

[@modestorosado1338]

When I realized that singularities were the reason behind the radius of convergence of Taylor series, I felt like I had been hit by a train. It blew my mind. This is one of the reasons why I find complex analysis so fascinating.

[@MrAlRats]

I'm dying to learn more! What I really want to know is what property does a function need to have in the neighbourhood of a point (in addition to being infinitely differentiable) to make it possible for its values to be approximated by a Taylor series?

[@modestorosado1338]

@@MrAlRats It depends on where you're doing analysis. The nicest set is of course the complex numbers, since there are a lot of conditions that are equivalent to analiticity. For example, if a function is holomorphic at a point, that is enough to ensure the existence of a Taylor expansion (around that point). Obviously this is not true for functions over the reals. As a matter of fact, there aren't any nice characterizations of analytic functions over the reals that I'm aware of. You can also look at analytic functions over the quaternions. Unfortunately, analiticity is a very restrictive condition in this case. If I recall correctly, not even linear functions over the quaternions are "quaternion" differentiable. Some are, but not all of them. In a sense, the reals are too small to see the whole picture and the quaternions are too big to be well-behaved. The sweet spot is the complex numbers.

[@angelmendez-rivera351]

Complex analysis is one of the most beautiful areas of mathematics.

[@hiltonmarquessantana8202]

This problem is beautifully discussed in the book: "Visual Complex Analysis".

[@triton62674]

@ ikr, could never be me

[@jurian0101]

+1

[@leif1075]

@ why too dry or dense and boring?

@@leif1075 i don't get the subject of your sentence: If the subject is "reading" then "i don't have an interior monologue, so it sucks for me", if the subject is me then "Yes"

[@giladzxc17]

This book is great. Im actually studying it right now

[@anonymousdude7982]

Me sitting here in my sophomore year of high school pretending like I understand this.

[@yimoawanardo]

You probably do 😃 I could swear age doesn't matter here. A minimum age would probably be 5 or 6, by the time you get the hang of talking basically lol Otherwise 40 or 12, you can understand even the "hardest" maths. My personal opinion though.

[@anonymousdude7982]

Yimo Awanardo That may just make me an idiot, but thank you. 🙂

[@mariogamesrock]

@@anonymousdude7982 dont worry, your not an idiot, I have no idea what kind of crack that dude is smoking. Until you have a fundamental understanding of basic calculus, which requires advanced algebra and trig, you (rightfully) should have no idea what a taylor series is. Just wait and your time will come

[@smrtfasizmu6161]

Well, you probably already know imaginary numbers, you will soon learn what derivative is and then you will learn Taylor series. Taylor series are just polynomials that approximate functions. They can approximate functions as close as you want them to (by having more and more terms in the polynomial), as long as the function that you want to approximate is analytical. The way you set up the polynomial is that you make sure that derivatives for some input of that polynomial match the derivatives for that same input of the function that you want to approximate. For instance, you set up the first derivative at x = 0 of your polynomial to be equal to the first derivative at x = 0 of the function that you want to approximate. Then add another term in the polynomial such that the second derivative at x = 0 is the same as second derivative at x=0 for the approximated function. And so on. I don't know what math curriculum is where you live, but it is possible that you will learn about derivatives next year. Then you can go back to this video, read this comment again and understand what this video is about.

[@yavuz1779]

Fake it till you make it

[@brboLikus]

Now it makes sense for it to be called the __radius__ of convergence. Because in 2D, it's kind of a misnomer.

[@angeldude101]

1D disks/balls exist and they have a radius and a surface. In fact, the intersection of a circle and a plane is exactly a 0-sphere, which is an object in 1D with a center and a radius, but only actually contains 2 points.

[@shashwattrivedi501]

One of the few channels whose content I watch regularly. Good job!

[@tomasstana5423]

Its nice that now we know ROC is connected to singularities in complex plane, but we still dont know why .... other than that, great video :)

[@zachstar]

Yeah this doesn't actually dive into the 'why' but that's because it is much more difficult to explain that, you have to dive further into complex analysis which is way beyond a video like this.

[@RC32Smiths01]

The ways in which imaginary numbers work in the real world never ceases to amaze. I think they will be pivotal to many more of life's advancements.

[@ΔημητρηςΜπεκιαρης-μ2κ]

All numbers are imaginary

[@pbj4184]

@@ΔημητρηςΜπεκιαρης-μ2κ And all numbers are real. Even the imaginary ones

[@clerklysquid6575]

Last time I was this early, pi hadn’t been calculated yet

[@henryginn7490]

technically it still hasn't, and won't ever be calculated in full

[@henryginn7490]

Demir Sezer I thought so, but the urge to point out a slight error overruled

[@-cookiezila-461]

Henry Ginn I don't think its an error, its purposely technically true

[@henryginn7490]

-COOKIEZILA - correct, I phrased it badly

[@KRYMauL]

Henry Ginn technically at 10^-34 you already calculated pi for all real world applications seeing as Physics breaks down at that point and we don’t know if what happens after that.

[@felixroux]

This guy's pfp is a pentagram and he has 666K subs at the moment.

[@hamiltonianpathondodecahed5236]

he is getting close to his mission of [redacted]

[@fernandobanda5734]

Not a pentagram tho

[@IMIGamerz2105]

It's now 667K subscribers.

[@mr.knight8967]

MATH QUESTION Numerical / algebra qué. kzbin.info/www/bejne/qaCxZoGjoMaLm68 One time see

[@shivangi296]

Beautiful! Thanks for the Mandelbrot mention. Guess your wallpaper with “imaginary” friends did a good job!

[@suyashverma15]

This was a total mind-blower, really! Would you like to make a video on fractals and its non integral dimensions also?

[@mrmoinn]

check out the video on it by 3Blue1Brown

[@suyashverma15]

@@mrmoinn Yes bro I have seen it and it was awesome too, but I am suggesting it to him too, because I think he can elaborate it on more of the practical side, with all its abstractness that Maths has to offer.

[@Lion1063]

We literally just went over Taylor/Maclaurin series in calc and I was so confused about the radius of convergence, this video was awesome, thanks

[@KillianDefaoite]

I'm taking a complex analysis course soon and I had never considered this. Thanks for the great video.

[@Saptarshi.Sarkar]

After 3 years of college Physics, I finally truly understand what radius of convergence means. Thanks.

[@VENOM-ol6pv]

Complex numbers are awesome! Thanks zachstar!☺

[@jonathangrey6354]

Even when not including complex numbers, I always assumed the RADIUS part meant all complex numbers within that radius of the center

[@hk8487]

best channel for engineers: Zack Star and for Mathematicians: 3B 1B

[@benjaminbrat3922]

Yes, I remember the dawning of understanding when I realized that the radius of convergence was actually... a radius... of convergence. (But in C)

[@mscir]

Thanks for this, I was an electronics tech, had to learn complex math but never understood how that played into things, only that it worked. Anything you do on complex numbers would be greatly appreciated.

[@technoguyx]

Very interesting. Never thought to ask myself if there's a deeper reason than the ratio test and even though I did take complex analysis last semester I never made the link. Love these visualizations

[@jamesbra4410]

Exception video! By far my favorite channel on KZbin. Keep up the good work. Perhaps you've readied your audience for Cauchy's Residue Theorem lol!

[@sandro7]

This was literally one of my biggest math questions for like a year or two, and I always figured it had to do with just something about the functions moving above and below the function without converging in it (like sinx doesn’t converge to 0), idk y I never thought of smth like this. The idea makes sense bc the derivatives won’t work out if it’s not analytic but I’m curious as to why the function can’t still be defined by the polynomial in other directions where the function is analytic (so the converging area isn’t just a circle).

[@royelhajj2612]

Beast like always Zach !!!

[@harrypotter5460]

Follow-up question for those with a curious mind: Is what Zach did for 1/(1+ x^2) always possible? More formally, is it always possible to extend a real analytic function (one with a Taylor series at every point) to a complex meromorphic function (one with a Taylor series at every point expect on a set of isolated poles) such that the radius of convergence of the Taylor series at a point is the distance from that point to the nearest complex pole? If so, is such an extension unique?

[@depressedguy9467]

I was 14 i knew about Fourier series but u was the guy to give me the intutive information about it

[@sarahrogers-pastio7709]

This is so cool! It's so wonderful finding things that make me fall even more in love with math gah

[@johnchessant3012]

Even more fun is how you can use the radius of convergence to find an asymptotic formula for the Maclaurin coefficients. I learned that in the book "generatingfunctionology" by Herbert Wilf.

[@DanteKG.]

Had a mental image of a dangerous place that fictional characters have to go to in order to fix something. "The machine is outputting anomalies with no reason! -We have to go down into the complex dimension to fix it.. -Why? What will we do there?? -Hunt for singularities"

[@daviddeane4250]

please make a video on mechatronics engineering and interdisciplinary fields

[@matiuspakpahan7612]

i wish you were my lecturer when i was in college

[@balconydemon4750]

this gave me goosebumps

[@Anna-n8n6n]

Great video

[@PhilipSmolen]

I love watching stuff I learned years ago, but explained with modern graphics!

[@MissTexZilla]

you can use imaginary numbers to calculate particle masses and fumble around with no boundary theories and apply those to either the general universe or stuff like blackholes.

[@yourmathtutorvids]

My face when it started moving at 5:45 😳 haha

[@yourmathtutorvids]

Shimmy Shai 😂 yes they did

[@geneeditor9545]

Great teaching. It helped me a lot to understand the topic.

[@denelson83]

That graph you introduce at 5:33 should have had its colours assigned based on the phase of the output.

[@kendakgifbancuher2047]

would like to see, how polynomial series "approach" that 3d plot at the end

[@anshumanagrawal346]

same

[@Arkunter]

Wow that was pretty nice.... I really enjoy these quality shorter videos!

[@chrisray1567]

Is there another step past imaginary numbers? Like quaternions? Do they describe 3-D spherical regions of convergence?

[@rafaelaassuncao9729]

dude that blew my mind

[@jurian0101]

A exceptionally fantastic case of poles ruin series expansion is the Sundman series of the 3-body astrodynamic system. The Sundman series is a CONVERGENT infinite series that solve arbitrary 3-body problem. Wait, what?! Per a PBS Spacetime episode, the catch is that in order to obtain that infinite series, a certain intricate manipulation on the complex plane is required - merely avoiding the poles (collisions of the bodies), which lead to an infinite series performing so poorly, converging only if 10^(N million) terms is added up. It is by no means a practical solution. :D (cf. Solving the Three Body Problem - PBS Spacetime)

[@alwaysdisputin9930]

6:27 Oh yeah! That's a fine pair of baps right there. Those real number only guys are missing out

[@samin3997]

This vedio helped me learn series solution of differential equations ❤️

[@perappelgren948]

Short video, but very to the point. 👍👍

[@johnginos6520]

Can you do a video on dynamics in social sciences, particularly economics, there has been work done on how gauge theory and differential geometry can be used in modeling economic issues

[@thephysicistcuber175]

Complex analysis

[@codewriter3000]

Make a tier list of every course you took in undergrad and grad school

[@Chemi4001]

this video makes me 　 Ｗ　Ｏ　Ｋ　Ｅ thanks, Zach Star

[@Callie_Cosmo]

As John Malani would say “You had me at ‘solved’”

[@charlescowan6121]

I had a shitty Calc 2 instructor who glossed over many sections. So when I got to upper division physics where series solutions were an expected skill, I really struggled! However when I studied complex analysis, I had a few very profound "Ahha" moments.

[@ilke3192]

I was like "why does this have only 120 views" then I realised I am very early... Wow

[@ifusubtomepewdiepiewillgiv1569]

i agre - i am making this comment to see how many comments are added by the time i reload this

[@princelumpypackmule1101]

Now it's at 120 x 10 views

[@necaton]

@@princelumpypackmule1101 Now it's at 120^10 views...or at least it should be

[@mathmachine4266]

(In reference to something he mentions at the end) Do removable singularities REALLY effect the radius of convergence? I mean, I have pretty much an unhealthy obsession with math, so, from my personal experience, under most circumstances, if there's a removable singularity, the rules of math will just pretend it isn't even there. So I'd be really interested to see a counter example to that behavior.

[@lateefahmadwanilaw8948]

Thank u sir

[@bernhardriemann3821]

hey man can you please tell the animation software you use for your videos?

[@zachstar]

For this one you can find the software used in the description :)

[@bernhardriemann3821]

@@zachstar thanks a billion

[@mindaugasdubosas7602]

Could you create a video for a general engineering major? What they do, jobs they can get etc? :)

[@taylorlorenztransormation3102]

I love your videos

[@michaeljaf5734]

You should make a video on construction engineering.

[@lolthistruebias5057]

Hey Zach! Can you maybe make a video about nuclear engineering?

[@varimas]

Can i ask what program you use to plot in 3D?

[@ahusky4498]

and this is why complex numbers fueled my passion for mathematics :D

[@physicsboi1744]

Hey Zach! What software do you use for maths animations?

[@zachstar]

In the description :)

[@physicsboi1744]

@@zachstar oh ty love ur videos :) Still remember when it was used to be MajorPrep

[@bee_irl]

Very good video

[@halim_Ha]

Is there any linear algebra in your channel

[@michaelupdike-bz6rg]

Do quaternions ever come up in a similar manner? Or even high dimensional numbers.

[@kingofgoldnessr9364]

I have the 3d graphing program he has but have no idea what to input to get that out, any tips?

[@nakodares5982]

One thing I don’t quite follow is that the function you showed doesn’t have a real output for every number in the complex plane. For example, x = 1 + i. So in this case, when it comes to that 3D plot of values for numbers in the complex plane, what would you plot as the output (z value)? It seems like your 3D plot was continuous over the complex plane, but then does that mean that the z value on your 3D plot wasn’t actually the value of the function? In which case what was it? I assumed it was because that’s what the y-axis is in 2D on the real number line. EDIT: I had a quick think and I assume you are just plotting the magnitude of the complex output. In which case I would guess that asymptotes in the complex plane remain asymptotes when you take the magnitude or something like that, in order to have the idea you presented about the radius of convergence hold up even when plotting the magnitude of the function.

[@zachstar]

Your edit is correct, I was just plotting the magnitude since that's all that was needed to show the singularities. I couldn't done phase with color but the program I was using doesn't seem to allow me to do that (I can only change color based on the z value)

[@nakodares5982]

@@zachstar Thanks for the reply and the video!

[@MozartJunior22]

I feel like this would be the best demonstration of complex numbers actually existing and being something you cannot ignore. In all other explanations, complex numbers are only a convenient addition. Also, I hate how an 8 minute video is now considered "shorter than usual"... I miss the old KZbin days where you can get your videos as bite sized knowledge. Nowadays each video is an f-ing documentary.

[@44xeon79]

Does that mean that a complex function would have a "sphere of convergence"?

[@jackskelotojack]

Is it just me or does this explain the powers in the fifth season in the way they can sess stuff out

[@HackersSun]

Is THAT what Taylor's series is used for? Oooooohh This is cool I can see the why

[@pabloariza2295]

woahhh thiis was rly cool

[@iTeerRex]

If you rotate this 5:35 by 90 degrees you get a much better picture ;-)

[@pebotin]

Very nice video..

[@simmikedia3347]

hey...how do you do such animation...these really intresting...wish even i could learn how to do...and your explanation are very clear ...keep going...all the best

[@zachstar]

The software I use is in the description :)

[@simmikedia3347]

@@zachstar Thank you😃😊

[@arjunarya2658]

5:37 reminds me of the Julia sets, probably has no connection though

[@SuperS953]

Is going to the complex numbers enough? Could an extension to quaternions or other bigger fields reveal more singularities?

[@angeldude101]

With regard to arithmetic closure, the complex numbers should be all that's needed as far as I'm aware. That said, there are alternatives that are mainly useful for different geometries, so the quaternions are best for representing rotations in 3D space, while the split-complex numbers are great for working with hyperbolic geometry.

[@demianpryima1015]

how do you find this stuff

[@fireblossom9618]

Was the title different before? I swear it was something about a calculus 2 question idk might've been something else

[@amineelfardi4311]

Can you please explain PID conlroller

[@tnk4me4]

I petition to have imaginary numbers renamed to something more intuitive like Topographic numbers or something.

[@angeldude101]

I personally prefer the term "spherical numbers" due to the intimate relationship they have with spherical geometry.

[@santi_z48]

This is the galaxy brain meme.

[@erickperez5930]

Hi guys, I need to plot some complex functions but I don´t know to program. Any software recommendations?

[@jamesjenkins9480]

Have you checked out manim, 3blue1brown's math animation library? I feel like that would be super useful for you.

[@furkansarihan]

Thank you...

[@Eyalkamitchi1]

Does this mean that the the Taylor series convergence over the imaginary plane? Because only then will the "convergence block" for the polynomial will be met. If so, how does it(Taylor series) do it(converge over the imaginary plane)? Or at least, why does it do that too, beside the real number line.

[@pierredonias8940]

But how do you find the radius of convergence in the first place?

[@Timothyday88]

I thought I was good at math until I saw this. Thank you

[@lexs7218]

for all calc 2 students who don’t understand why when and how the Taylor series actually converges to the function as to as more and more terms,…. well, well, well,….. i could answer that question but then you wouldn’t suffer like i did staring at wikipedia pages of Cauchy integral formula looking things

[@blinded6502]

Okay But why? Why are these complex number singularities relevant, if we never use them and only deal with real numbers?

[@vazmeee]

Hi zach. Could you help us understand the beta binomial distributions anytime exploring all possible ranges of alpha and beta in a intuitive way. I've always had some trouble wrapping my head around when it comes to values

[@mepkn]

Every number is a Complex Number, the earlier the people know it, the easier the math becomes.

[@PapaFlammy69]

111111

[@technoguyx]

reported for hate speech

[@dirichlettt]

Papi

[@ΑντώνηςΝικολακάκης-ω2γ]

Me studying 1-1 functions, sees this. Me. Exe stopped working

[@danielironside815]

Everybody:"Damn, this is such an interesting topic" Me: Hehe boobies 5:37

[@-cookiezila-461]

No. I must be imagining this.

[@theunknown4834]

Try using geogebra instead of desmos