The square of a formal laurent series

Thread starter R.P.F.
Start date Oct 16, 2011
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Laurent series Series Square

In summary, a formal Laurent series is a power series that includes both positive and negative powers of the variable and may not necessarily converge for all values of x. The square of a formal Laurent series refers to the result of multiplying the series by itself, and has various applications in mathematics. To calculate the square of a formal Laurent series, one can use the Cauchy product formula or the binomial theorem. The convergence of the square of a formal Laurent series is determined by its coefficients and may not converge for all values of x, but if the original series converges absolutely, then its square will also converge absolutely.

Oct 16, 2011

R.P.F.

Let [tex]F[/tex] be a field. Let [tex] c \in F[/tex].

I am trying to show that if
[tex] c = f^2[/tex] where [tex]f\in F[[x]][/tex], then [tex]f\in F[/tex].

So I am able to get rid of the terms with negative exponent. So now I'm left with a formal power series. Anyone knows how to do this? Thanks!

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Oct 16, 2011

R.P.F.

Figured it out. So sorry, guys. :(

What is a formal Laurent series?

A formal Laurent series is a power series that includes both positive and negative powers of the variable, such as f(x) = ∑_n=-∞∞ a_nxⁿ. It is considered a formal series because it may not necessarily converge for all values of x.

What is the square of a formal Laurent series?

The square of a formal Laurent series refers to the result of multiplying the series by itself, resulting in a new series with coefficients that are the product of the original series' coefficients. For example, if f(x) = ∑_n=-∞∞ a_nxⁿ, then f(x)^2 = ∑_n=-∞∞ (∑_k=-∞∞ a_kx^k)(∑_m=-∞∞ a_mx^m) = ∑_k=-∞∞ (∑_m=-∞∞ a_k-ma_m)x^k.

What is the significance of the square of a formal Laurent series?

The square of a formal Laurent series has various applications in mathematics, particularly in complex analysis. It can be used to find the inverse of a Laurent series, as well as to determine the convergence of a series. It also has connections to other mathematical concepts, such as generating functions and partition functions.

How is the square of a formal Laurent series calculated?

To calculate the square of a formal Laurent series, you can use the Cauchy product formula, which states that the coefficient of xⁿ in the product of two power series is equal to the sum of the products of the coefficients of x^k in the first series and x^n-k in the second series, where k ranges from -∞ to ∞. Alternatively, you can also use the binomial theorem to expand the square of a formal Laurent series into a power series.

Can the square of a formal Laurent series diverge?

Yes, the square of a formal Laurent series may not converge for all values of x. This is because the convergence of a series is determined by its coefficients, and multiplying the coefficients by themselves may result in a series with different convergence properties. However, if the original series converges absolutely, then its square will also converge absolutely.