(Though, it wouldn't surprise me if there's a clever way to do this that gives you a closed form)Hurkyl said:I don't see the difficulty in multiplying the series... I guess you don't like the fact that each coefficient will be an infinite sum.
A Laurent series is a type of mathematical series that represents a complex function as an infinite sum of terms. It is commonly used to study functions in complex analysis.
The function f(z)=1/z^2 is a complex function that maps every point in the complex plane to its reciprocal squared. This means that for any complex number z, f(z) is equal to 1 divided by z squared.
This notation indicates that the Laurent series is being evaluated at a point a, which is outside the circle of convergence (i.e. |z-a|>|a|). In other words, the series is being evaluated at a point where the function is not analytic.
To find the Laurent series of f(z)=1/z^2 for |z-a|>|a|, you can use the method of partial fractions. First, express the function as a sum of two terms (1/z and 1/z^2), and then expand each term using their respective power series. The resulting series will be the Laurent series of f(z)=1/z^2 for |z-a|>|a|.
The Laurent series of f(z)=1/z^2 for |z-a|>|a| is important because it allows us to study the behavior of the function at points where it is not analytic. This can provide insights into the singularities and poles of the function, and how it behaves near these points.