Hey, Leo
Yes, absolutely they are interchangeable. The idea of using a multivalued function or a Riemann surface. My point is simply, it's streamlined if you use Riemann surfaces. By which, I mean, certainly it's equivalent; but it'll be a bit slow moving when you want to prove grand things about this theory. I only mean to say it as it's a better choice of language. Using the language of Riemann surfaces will open many doors; especially if you're talking about multi-valued functions. Where it can be kind of stiff; where as with Riemann surfaces, the language is more fluid. That's all I mean.
Oh, and don't worry about your language
Everything's fine; I apologize if my tone came off a bit blunt. I didn't mean to sound like that. And you speak very well, so you have nothing to apologize about. I understand that English is my first language, and therefore, I instantly have an advantage.
Everything you are saying is very well thought out mathematically; I'd hate for you to feel that I'm discouraging you. I'm very impressed by everything here. I'm simply making the point that this seems more at home with Riemann surfaces, is all; especially if you want to describe all the multivalued branches. It'll just appear more naturally in this language (at least, I feel).
But by all means, keep at it. This is fantastic stuff!
Regards, James.
PS: Good luck on your exam!
Yes, absolutely they are interchangeable. The idea of using a multivalued function or a Riemann surface. My point is simply, it's streamlined if you use Riemann surfaces. By which, I mean, certainly it's equivalent; but it'll be a bit slow moving when you want to prove grand things about this theory. I only mean to say it as it's a better choice of language. Using the language of Riemann surfaces will open many doors; especially if you're talking about multi-valued functions. Where it can be kind of stiff; where as with Riemann surfaces, the language is more fluid. That's all I mean.
Oh, and don't worry about your language
Everything's fine; I apologize if my tone came off a bit blunt. I didn't mean to sound like that. And you speak very well, so you have nothing to apologize about. I understand that English is my first language, and therefore, I instantly have an advantage. Everything you are saying is very well thought out mathematically; I'd hate for you to feel that I'm discouraging you. I'm very impressed by everything here. I'm simply making the point that this seems more at home with Riemann surfaces, is all; especially if you want to describe all the multivalued branches. It'll just appear more naturally in this language (at least, I feel).
But by all means, keep at it. This is fantastic stuff!
Regards, James.
PS: Good luck on your exam!