I'm not an expert on the history here, but I should remark that I've reversed the historical development of the subject as is typical now. The original objects of interest were elliptic integrals (see the following video) which showed up in computing the arc length of ellipses. Abel and Jacobi came across the idea of inverting these functions which were then observed to be doubly periodic. Their inspiration came from the fact that similar (but simpler) integrals gave rise to inverse trigonometric functions so they wondered if the inverse functions to these elliptic integrals were more natural to work with. Now, if you start instead from the viewpoint of trying to construct doubly periodic functions, the p-functions are rather natural and I guess Weierstrass tried to work out how to go from there to the theory of elliptic integrals.
Very interesting and clear lesson, but I find a little weird that the very first example does not satify the definition perfectly (it is not defined on all C).
Sorry, I'm going off-topic but I just had an idea pop in my head. I want to find the functions for vector . x,y,z are real numbers, n is a whole number. This vector is an ellipsoid based on the ellipse x^2±x*y+y^2=1. Sorry, this intuitively makes sense to me, but probably not to you. I only finished grade 12 pre-calculus. Anyway, if I stepped it down to and made x a complex number and y a real number, the functions should be solvable. If it isn't, that would be damaging to complex numbers as a whole, wouldn't it?
Very clear exposition! Thank you so much :)
Brilliantly explained
I did enjoy this adventure in pure mathematics.
this is a great video, thank you so much
Quite astonishing. I wonder how Weierstrass figured this out originally, and why?
I'm not an expert on the history here, but I should remark that I've reversed the historical development of the subject as is typical now. The original objects of interest were elliptic integrals (see the following video) which showed up in computing the arc length of ellipses. Abel and Jacobi came across the idea of inverting these functions which were then observed to be doubly periodic. Their inspiration came from the fact that similar (but simpler) integrals gave rise to inverse trigonometric functions so they wondered if the inverse functions to these elliptic integrals were more natural to work with. Now, if you start instead from the viewpoint of trying to construct doubly periodic functions, the p-functions are rather natural and I guess Weierstrass tried to work out how to go from there to the theory of elliptic integrals.
Understandably explained. It is cool.
Woah that's so cool
Amazing!
Thank you for teaching me and thank you
Very interesting and clear lesson, but I find a little weird that the very first example does not satify the definition perfectly (it is not defined on all C).
I think all non-constant doubly-periodic meromorphic functions have poles? By the argument he mentioned about Liouville's theorem.
Very succinct, Thanks!
Man U need more subscribers. Amazing video, if you can could you please make more videos on p adic numbers
Thanks for your comments. I hope to add a couple more videos to the playlist on p-adic numbers and Diophantine equations.
thanks a whole lot
Wow 🤩
Very good.
Sorry, I'm going off-topic but I just had an idea pop in my head. I want to find the functions for vector . x,y,z are real numbers, n is a whole number. This vector is an ellipsoid based on the ellipse x^2±x*y+y^2=1. Sorry, this intuitively makes sense to me, but probably not to you. I only finished grade 12 pre-calculus. Anyway, if I stepped it down to and made x a complex number and y a real number, the functions should be solvable. If it isn't, that would be damaging to complex numbers as a whole, wouldn't it?
You should say Laurent series not Taylor series :)