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?
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)
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.
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?
@@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.
@@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"
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.
@@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.
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.
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 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
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.
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.
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.
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
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).
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?
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.
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
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.
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)
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"
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.
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)
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.
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
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.
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.
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
(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.
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 UA-cam days where you can get your videos as bite sized knowledge. Nowadays each video is an f-ing documentary.
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
Does anyone have any tips for a student having to learn calc1, calc2 and calc3 (about half of calc2 and 3) in about 4 weeks? Asking ofcourse for a friend :P
@@fernandobanda5734 definitely for a friend and NOT for my university programme with teachers that don't know how technology works and end up saying "Well, it's in the book so go read that"
What a coincidence, we were just learning about Maclaurin series at school today. How can you "centre" a series around a point other than zero? Is this something to do with Taylor series?
Yeah, that's exactly it! Taylor series are simply a generalisation of Maclaurin series. Instead of having the n-th derivative at zero and then x^n as the terms in your sum, you simply take the n-th derivative at the point y where you want to centre the series and then multiply by (x-y)^n instead of just x^n. (The 1/n! stays the same). Interestingly, as this is nothing but a shift of the argument of the function, to prove that Taylor series in this sense exist requires nothing more than to prove that Maclaurin series exist. Hope that was somewhat illuminating!
Essentially you only do a bit of renaming. Say you want to do the series at x=1. Then you introduce a z so that z=x-1 and transform your Formula. Now you notice that your x=1 conveniently is at z=0, say to yourself " x, z, what's a name anyway" and calculate for z=0 like you would for x=0. Afterwards you rename again and replace every z by x-1. So e.g. z² becomes (x-1)². That's it!
For me calc 1 was limits, derivatives, integrals for the first time (related rates, optimization, volume of revolution, etc). Calc 2 was integration techniques (by parts, trig sub, etc), then series/sequences and Taylor/maclaurin series Calc 3 is multi variable calculus where you first learn the partial derivative, 3d graphs, double and triple integrals, and a little vector analysis. After that you enter into the courses that we all know by name like differential equations, linear algebra, real analysis, and so on.
@@dukeofworcestershire7042 There are high school students who complete all three of the calc courses but it's not the majority (my school only offered up to calc 2 actually). Plenty take calc 1 but after that there's a big drop in terms of who goes to calc 2.
@@zachstar I see. Where I live it's a bit more standardized, all students take the same math courses. I'm what I think is equivalent to a highschool freshman and have been getting into math lately, but because the vast majority of good content is English there is a certain cultural barrier as some stuff doesn't translate well between school systems, thus making it difficult to figure out where to start
New STEMerch Store: stemerch.com/
stupid people: ?
genius people: { }
Please make a video on what is mechatronics and its future
MATH QUESTION
Numerical / algebra qué.
ua-cam.com/video/soN5NmkaXeM/v-deo.html
One time see
im confused, for stability sigma shud be negative decaying , but region of convergence says it shud be positive . whats happening?
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?
Imaginary numbers? Jeez when are we ever gunna use this stuff!?
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)
i and j are your best friends in college....
@@phillipgrunkin8050 :)
@@AndrewDotsonvideos Nice to see you on one of Zachs videos.
Imagine not being able to Wick rotate.
That also explains why it's called a "radius" of convergence instead of just a region of convergence. Cool!
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.
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?
@@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.
Complex analysis is one of the most beautiful areas of mathematics.
This problem is beautifully discussed in the book: "Visual Complex Analysis".
@ ikr, could never be me
+1
@ 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"
This book is great. Im actually studying it right now
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.
All numbers are imaginary
@@ΔημητρηςΜπεκιαρης-μ2κ And all numbers are real. Even the imaginary ones
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 :)
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.
One of the few channels whose content I watch regularly. Good job!
Beautiful! Thanks for the Mandelbrot mention. Guess your wallpaper with “imaginary” friends did a good job!
I'm taking a complex analysis course soon and I had never considered this. Thanks for the great video.
This was a total mind-blower, really! Would you like to make a video on fractals and its non integral dimensions also?
check out the video on it by 3Blue1Brown
@@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.
Last time I was this early, pi hadn’t been calculated yet
technically it still hasn't, and won't ever be calculated in full
Demir Sezer I thought so, but the urge to point out a slight error overruled
Henry Ginn I don't think its an error, its purposely technically true
-COOKIEZILA - correct, I phrased it badly
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.
Me sitting here in my sophomore year of high school pretending like I understand this.
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.
Yimo Awanardo That may just make me an idiot, but thank you. 🙂
@@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
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.
Fake it till you make it
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
This guy's pfp is a pentagram and he has 666K subs at the moment.
he is getting close to his mission of [redacted]
Not a pentagram tho
It's now 667K subscribers.
MATH QUESTION
Numerical / algebra qué.
ua-cam.com/video/soN5NmkaXeM/v-deo.html
One time see
Complex numbers are awesome!
Thanks zachstar!☺
Now it makes sense for it to be called the __radius__ of convergence. Because in 2D, it's kind of a misnomer.
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.
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.
Exception video! By far my favorite channel on UA-cam. Keep up the good work. Perhaps you've readied your audience for Cauchy's Residue Theorem lol!
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
After 3 years of college Physics, I finally truly understand what radius of convergence means. Thanks.
I was 14 i knew about Fourier series but u was the guy to give me the intutive information about it
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).
Even when not including complex numbers, I always assumed the RADIUS part meant all complex numbers within that radius of the center
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?
please make a video on mechatronics engineering and interdisciplinary fields
This is so cool! It's so wonderful finding things that make me fall even more in love with math gah
best channel for engineers: Zack Star
and for Mathematicians: 3B 1B
Beast like always Zach !!!
I love watching stuff I learned years ago, but explained with modern graphics!
would like to see, how polynomial series "approach" that 3d plot at the end
same
i wish you were my lecturer when i was in college
Yes, I remember the dawning of understanding when I realized that the radius of convergence was actually... a radius... of convergence. (But in C)
this gave me goosebumps
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.
Wow that was pretty nice.... I really enjoy these quality shorter videos!
This vedio helped me learn series solution of differential equations ❤️
Great teaching. It helped me a lot to understand the topic.
dude that blew my mind
Great video
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
That graph you introduce at 5:33 should have had its colours assigned based on the phase of the output.
this video makes me Ｗ Ｏ Ｋ Ｅ
thanks, Zach Star
Short video, but very to the point. 👍👍
Complex analysis
Make a tier list of every course you took in undergrad and grad school
Is there another step past imaginary numbers? Like quaternions? Do they describe 3-D spherical regions of convergence?
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.
Could you create a video for a general engineering major? What they do, jobs they can get etc? :)
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)
6:27 Oh yeah! That's a fine pair of baps right there. Those real number only guys are missing out
As John Malani would say
“You had me at ‘solved’”
Hey Zach! Can you maybe make a video about nuclear engineering?
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"
Thank u sir
You should make a video on construction engineering.
Do quaternions ever come up in a similar manner? Or even high dimensional numbers.
I love your videos
Very good video
My face when it started moving at 5:45 😳 haha
Shimmy Shai 😂 yes they did
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.
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)
@@zachstar Thanks for the reply and the video!
Can you please explain PID conlroller
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.
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
The software I use is in the description :)
@@zachstar Thank you😃😊
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.
Is going to the complex numbers enough? Could an extension to quaternions or other bigger fields reveal more singularities?
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.
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
woahhh thiis was rly cool
(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.
Very nice video..
Is there any linear algebra in your channel
Thank you...
Was the title different before? I swear it was something about a calculus 2 question idk might've been something else
Is THAT what Taylor's series is used for?
Oooooohh
This is cool I can see the why
I have the 3d graphing program he has but have no idea what to input to get that out, any tips?
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 UA-cam days where you can get your videos as bite sized knowledge. Nowadays each video is an f-ing documentary.
hey man can you please tell the animation software you use for your videos?
For this one you can find the software used in the description :)
@@zachstar thanks a billion
Is it just me or does this explain the powers in the fifth season in the way they can sess stuff out
I was like "why does this have only 120 views" then I realised I am very early... Wow
i agre - i am making this comment to see how many comments are added by the time i reload this
Now it's at 120 x 10 views
@@princelumpypackmule1101 Now it's at 120^10 views...or at least it should be
i want to learn & understand mathematics like you. how can i learn? please suggest me book or tutorials for mathematics from basic.
Can i ask what program you use to plot in 3D?
Have you checked out manim, 3blue1brown's math animation library? I feel like that would be super useful for you.
Hey Zach! What software do you use for maths animations?
In the description :)
@@zachstar oh ty love ur videos :) Still remember when it was used to be MajorPrep
5:37 reminds me of the Julia sets, probably has no connection though
If you rotate this 5:35 by 90 degrees you get a much better picture ;-)
Does that mean that a complex function would have a "sphere of convergence"?
Hi guys, I need to plot some complex functions but I don´t know to program. Any software recommendations?
how do you find this stuff
and this is why complex numbers fueled my passion for mathematics :D
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
Hey what tool did he use to graph complex nos in 3d? Anyone?
In the description!
Does anyone have any tips for a student having to learn calc1, calc2 and calc3 (about half of calc2 and 3) in about 4 weeks? Asking ofcourse for a friend :P
I have a question: why?
@@fernandobanda5734 definitely for a friend and NOT for my university programme with teachers that don't know how technology works and end up saying "Well, it's in the book so go read that"
@@rednassie1101 It's pretty crazy to learn that many topics that fast, even if they weren't complicated.
@@rednassie1101 Sorry if I don't have useful tips. Just concentrate and practice, I guess. :/
@@fernandobanda5734 haha, thanks for the heads up. Have a good day
My precalc *ss just trying to keep along! XD
I thought I was good at math until I saw this. Thank you
Awesome
What a coincidence, we were just learning about Maclaurin series at school today.
How can you "centre" a series around a point other than zero? Is this something to do with Taylor series?
Yeah, that's exactly it! Taylor series are simply a generalisation of Maclaurin series. Instead of having the n-th derivative at zero and then x^n as the terms in your sum, you simply take the n-th derivative at the point y where you want to centre the series and then multiply by (x-y)^n instead of just x^n. (The 1/n! stays the same). Interestingly, as this is nothing but a shift of the argument of the function, to prove that Taylor series in this sense exist requires nothing more than to prove that Maclaurin series exist. Hope that was somewhat illuminating!
Essentially you only do a bit of renaming.
Say you want to do the series at x=1. Then you introduce a z so that z=x-1 and transform your Formula. Now you notice that your x=1 conveniently is at z=0, say to yourself " x, z, what's a name anyway" and calculate for z=0 like you would for x=0. Afterwards you rename again and replace every z by x-1. So e.g. z² becomes (x-1)².
That's it!
Pretty elegant explanations, thanks! It makes this video fit together much more nicely.
But how do you find the radius of convergence in the first place?
Me studying 1-1 functions, sees this.
Me. Exe stopped working
Can someone explain to me how the divisions in fields of maths in the American school system work? (Calculus 1, Calculus 2 etc.)
For me calc 1 was limits, derivatives, integrals for the first time (related rates, optimization, volume of revolution, etc).
Calc 2 was integration techniques (by parts, trig sub, etc), then series/sequences and Taylor/maclaurin series
Calc 3 is multi variable calculus where you first learn the partial derivative, 3d graphs, double and triple integrals, and a little vector analysis.
After that you enter into the courses that we all know by name like differential equations, linear algebra, real analysis, and so on.
@@zachstar I see! Is that stuff covered in high school or is it a university thing?
@@dukeofworcestershire7042 There are high school students who complete all three of the calc courses but it's not the majority (my school only offered up to calc 2 actually). Plenty take calc 1 but after that there's a big drop in terms of who goes to calc 2.
@@zachstar I see. Where I live it's a bit more standardized, all students take the same math courses. I'm what I think is equivalent to a highschool freshman and have been getting into math lately, but because the vast majority of good content is English there is a certain cultural barrier as some stuff doesn't translate well between school systems, thus making it difficult to figure out where to start
Try using geogebra instead of desmos