- #1

the_pulp

- 207

- 9

- TL;DR Summary
- Is there a kind of "Laurent Serie" for each algebraic function over it´s whole Riemann Surface? Perhaps using radicals in the approximation?

Hi, I'm writting because I sort of had an idea that looks that it should work but, I did not find any paper talking about it. I was thinking about approximating something like algebraic functions. That is to say, a function of a complex variable z,(probably multivalued) that obeys something like:

p

I know that given that this is probably multivalued, we have to impose some kind of branch cut and, as a consequence, in general, we can not generate a laurent series using the general formula to obtain its coefficients:

a

In general, what happens is that there is some kind of issue when the loop crosses the branch cut given that when it does, the function does not return to its original value. However, what I thought is that, for example (just to simplify the explanation), for a "bivalued" algebraic function, given that after 2 loops, the function returns to its original value, something like this should work:

a

y(z) = ∑a

(note that give the "n/2" in the exponent of z, we are approximating the "bivalued" function "y(z)" with "bivalued" "monomials" -so we have apples and apples in both sides of the equation-)

I find that something like these should be true because when you express z as c + r*e

This idea sounds that it should work with three valued algebraic functions (using a curve γ that performs three loops and dividing the exponent by 3) or with, in general, "n" valued algebraic functions (using a curve γ that performs "n" loops and dividing the exponent by "n").

Even though these ideas sounds feasible, I do not find anything on the web that describes this setting (the closest thing that I found is something like puseaux series, but it is not estimated, in general with the philosophy used in general in laurent series). Given that there is a negligible probability that this setting has not been studied in the past, I'm asking you guys:

Have you seen any kind of approximation of algebraic function overs its whole Riemann Surface similar to what I'm imagining? (not just local approximations of just one branch)

If not, do you find any hole in the argument, in order to grasp some kind of intuition of why this does not exist?

Thanks in advance for you usual kindful help.

Ps: Sorry, I did not find the mathematical equation writer in order to be more clear. In addition, I'm not an native English speaker so, sorry for any mistake in my explanation. Finally, I'm sorry also for not being precise in my explanation. Given that what I have in mind is an intuition, not a mathematical theorem, this is what you guys are receiving.

p

_{1}(z)*y^n + p_{2}(z)*y^(n-1) + ... + p_{n+1}(z)*y^0 = 0I know that given that this is probably multivalued, we have to impose some kind of branch cut and, as a consequence, in general, we can not generate a laurent series using the general formula to obtain its coefficients:

a

_{n}=1/(2*pi*i)* ∫_{γ}y(z)/(z-c)^(n+1) * dzIn general, what happens is that there is some kind of issue when the loop crosses the branch cut given that when it does, the function does not return to its original value. However, what I thought is that, for example (just to simplify the explanation), for a "bivalued" algebraic function, given that after 2 loops, the function returns to its original value, something like this should work:

a

_{n/2}=1/(**4***pi*i)* ∫_{γ}y(z)/(z-c)^(n**/2**+1) * dzy(z) = ∑a

_{n/2}*z^{n/2}**In this case γ should be a curve that goes around c two times.**(note that give the "n/2" in the exponent of z, we are approximating the "bivalued" function "y(z)" with "bivalued" "monomials" -so we have apples and apples in both sides of the equation-)

I find that something like these should be true because when you express z as c + r*e

^{θ*i}then you sort of have all the functions that a Fourier series should have in order to approximate any periodic function (in coloquial terms, θ, given that is divided by 2, goes half as fast as in the usual laurent series, and, as a consequence, when you go around the 2 loops of γ, θ/2 performs just one loop so, we go back to the usual laurent series.This idea sounds that it should work with three valued algebraic functions (using a curve γ that performs three loops and dividing the exponent by 3) or with, in general, "n" valued algebraic functions (using a curve γ that performs "n" loops and dividing the exponent by "n").

Even though these ideas sounds feasible, I do not find anything on the web that describes this setting (the closest thing that I found is something like puseaux series, but it is not estimated, in general with the philosophy used in general in laurent series). Given that there is a negligible probability that this setting has not been studied in the past, I'm asking you guys:

Have you seen any kind of approximation of algebraic function overs its whole Riemann Surface similar to what I'm imagining? (not just local approximations of just one branch)

If not, do you find any hole in the argument, in order to grasp some kind of intuition of why this does not exist?

Thanks in advance for you usual kindful help.

Ps: Sorry, I did not find the mathematical equation writer in order to be more clear. In addition, I'm not an native English speaker so, sorry for any mistake in my explanation. Finally, I'm sorry also for not being precise in my explanation. Given that what I have in mind is an intuition, not a mathematical theorem, this is what you guys are receiving.

Last edited: