@xenontesla122
That also explains why it's called a "radius" of convergence instead of just a region of convergence. Cool!
@AndrewDotsonvideos
Imaginary numbers? Jeez when are we ever gunna use this stuff!?
@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.
@hiltonmarquessantana8202
This problem is beautifully discussed in the book: "Visual Complex Analysis".
@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.
@shivangi296
Beautiful! Thanks for the Mandelbrot mention. Guess your wallpaper with “imaginary” friends did a good job!
@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 :)
@KillianDefaoite
I'm taking a complex analysis course soon and I had never considered this. Thanks for the great video.
@suyashverma15
This was a total mind-blower, really! Would you like to make a video on fractals and its non integral dimensions also?
@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
@brboLikus
Now it makes sense for it to be called the _radius_ of convergence. Because in 2D, it's kind of a misnomer.
@anonymousdude7982
Me sitting here in my sophomore year of high school pretending like I understand this.
@felixroux
This guy's pfp is a pentagram and he has 666K subs at the moment.
@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.
@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?
@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).
@supremum24
Even when not including complex numbers, I always assumed the RADIUS part meant all complex numbers within that radius of the center
@Saptarshi.Sarkar
After 3 years of college Physics, I finally truly understand what radius of convergence means. Thanks.
@zachstar