Complex analysis: Roots

This lecture is part of an online undergraduate course on complex analysis.
We show how to express mutliplication of complex numbers using polar coordinates. As applications we show that complex numbers all have n'th roots, and derive the formulas for sin and cos of multiple angles.
For the other lectures in the course see • Complex analysis

[@JaybeePenaflor]

Loving this series as taught by a Fields medal recipient!!! Thanks, Dr Borcherds!!!

[@caspermadlener4191]

There is a "best" way to visualise complex multiplication, since the triangle with points 0, 1 and z₁ is similar to the triangle with points 0, z₂ and z₁z₂, uniquely determining z₁z₂, as long as you use the right type of similarity of not requiring reflection, only scaling and rotation.
This similarity has the nice property that SAB ~ SCD ⇔ SAC ~ SBD.

[@willtyler3093]

Very insignificant correction: at 30:45 it should be i(sinx)^5 not 5i(sinx)^5. Loving this series, it's providing a much needed refresher of complex analysis for me (and I suspect many others). I took complex analysis in my undergrad, and it stopped as we reached cauchy's residue theorem - I'm hopeful this series will extend somewhat further.

[@barbietripping]

I'm excited that I've understood the beginning of this course from previous study, but I'm nervous you'll soon upload a lesson that will humble me

[@arturjorge5607]

I feel the same

[@serioussearch9383]

Every mathematical book that is worth reading must be read "backwards and forwards," if I may use the expression. I would modify Lagrange's advice a little and say, "Go on, but often return to strengthen your faith." When you come to a hard or dreary passage, pass it over; and come back to it after you have seen its importance or found the need for it further on.
- George Chrystal

[@danielprovder]

@Serious Search, I took a full year high-level ug abstract algebra cource and by the end of it I noted that one could proceed logically working backwards, the prof had no comment.

[@bendavis2234]

@@serioussearch9383 That's an awesome and very useful quote

[@INT41O]

@@serioussearch9383 This is true, since mathematical books are usually badly written.

[@orthoplex64]

Thanks for the video Dr. Borcherds :)
At 9:40 you mention quaternions are used for rotations in computer systems for efficiency. I want to add that another nice benefit is that interpolating between rotations as quaternions, by spherical linear interpolation on S³, works well and is much simpler than with matrices.

[@jacobolus]

Note that most of the time it is atan2(y, x), with y as the first argument. If you want to implement this and you only have one-argument atan available, you can use the inverse stereographic projection (half-angle tangent), atan2 = (y, x) => 2 * atan(y / (x + sqrt(x*x + y*y)))

[@stockdaleroger]

It's a brilliant course! It's straightforward for the essence of complex analysis.

[@idolgin776]

To me this is the most fascinating topic in Mathematics. So much fun to learn and teach!

[@evionlast]

Phenomenal

[@adk3141]

At 14:33 I think almost all computer languages define atan2(y,x) (rather than atan2(x,y)) as being the argument of x + iy. Sorry to nit-pick!

[@joelcurtis562]

Excellent review of complex numbers. Enjoying the series, thank you.

[@putin_navsegda6487]

I enjoy your lectures! Thank you sir Richard E Borcherds!

[@barbietripping]

Is it suitable to say Theta = arctan(y/x) + (pi/2) (1 - x/|x|) as long as x is not zero?
This way x/|x| gets the sign of the real part of the complex number, and (1-x/|x|) is either 0 or 2.

[@paolofernandofloresrivera6244]

the best teacher , greetings from mexico

[@Pterodactyl42]

viva la razaaaa

[@Pterodactyl42]

question: when prof. Richard explains that we can’t prove continuity for square roots of complex numbers at 21:00 does this mean the values we used for reference literally changed? how does this have any bearing on, let’s say, our physical reality?
I appreciate answers!

[@gunhasirac]

The quick answer is there’s no “real meaning” of complex-valued square root function because it’s defined by imaginary numbers.
Detailed answer will be “domain of complex-valued square root function is not the complex plane”. Instead the domain is the so-called Riemann surface, which is 2 copies of complex plane glued together without intersecting itself. If you try to look this up it does intersect itself reason being it cannot be embedded into 3-dimensional space. There’s visualization available on TH-cam if you really want to know what is happening. Roughly speaking, as mentioned in the lecture, any continuous curve goes around the origin once back to the starting point will be mapped to a curve that only goes to the opposite side of the starting point. To make the image of a curve into a loop ( a curve that starts and ends at the same point ), simply goes around the origin twice will do. This is why we need 2 copies of complex plane as well.

[@loveihcnurnat664]

I'm confused why it's stupid to not choose 1 as the square root of itself? And why we wish to make it continuous? If we define the square root of (x+yi) where y

[@ldb579932]

Correction. At 22:50 you offered an example 5th root of 4 + 3i. Once you represented it on the complex plane, you called and labeled it 3 + 4i. Based on the scale provided by the unit circle, it looks more like 4 + 2i.
I realize this mistake has no bearing on the main point of how 5th roots are computed and located on the plane, but when will I ever have a chance to correct someone of such mathematical stature? I would bow down to you in your presence.

[@oni8337]

28:20 is de moivre's theorem?

[@pmcate2]

@6:05 Doesn't he mean S2? A circle is a 2d sphere

[@T-qx74]

Mr Borcherds is all class

[@migarsormrapophis2755]

ye

[@paul_tee]

widepeepohappy