Power Series Expansion of Dedekind Eta Function: How to Expand η(τ)/η(3τ)?

In summary, a power series expansion is a way to represent a function as an infinite sum of terms, each of which is a constant multiple of a power of the independent variable. It is useful for approximating complicated functions and evaluating them at specific points. The coefficients in a power series expansion can be found using the Taylor series formula, the method of undetermined coefficients, or by comparing known series. The radius of convergence is the maximum distance from the center where the series still converges and can be determined using the ratio test. However, not all functions can be represented by a power series expansion, as they must be analytic and have a finite radius of convergence.
  • #1
twoforone
3
0
Hi Every body!

I wan to compute the power series expansion of dedekind eta function. Specifically, I want to know the power series expansion of η(τ)/η(3τ)? How could I expand this function? I would be happy if you could help me as I am stuck at this state when I am computing the modular polynomial of prime number 3.
 
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  • #2
is eta the alternating zeta function??
 
  • #3
eta function is a function with this expression
infinity
η(τ)=q^24 ∏(1-q^n)
n=0
where q=e^2πiτ
Sorry, the product is from n=0 up-to infinity. You should understand in that way.
 

What is a power series expansion?

A power series expansion is a representation of a function as an infinite sum of terms, each of which is a constant multiple of a power of the independent variable. It is used to approximate a function and to evaluate it at a specific point.

Why is a power series expansion useful?

A power series expansion allows us to approximate complicated functions using simpler polynomials. This can make calculations and analyses easier and more efficient, especially for functions that do not have a simple closed form expression.

How do you find the coefficients in a power series expansion?

The coefficients in a power series expansion can be found by using the Taylor series formula, which involves taking derivatives of the function at a specific point. Alternatively, they can also be found using the method of undetermined coefficients or by comparing the power series to known series.

What is the radius of convergence in a power series expansion?

The radius of convergence is the distance from the center of the power series where the series converges. It represents the maximum distance from the center at which the series can still converge. The radius of convergence can be determined by using the ratio test on the coefficients of the power series.

Can a power series expansion be used for any function?

No, a power series expansion can only be used for functions that are analytic, meaning they have derivatives of all orders. Additionally, the function must have a finite radius of convergence for the power series to be a valid representation of the function.

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