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Posted on November 20, 2014
The holiday season can bring some of the worst traffic around. We already know that more people are hitting the road to visit their families during the holiday season. This is great if they are going to visit family in another city or region. But many people are driving to the destination that is on their holiday wish list. This can be a great thing but it can also become a nightmare. On this episode I will show you how you can better your chances of beating the holiday traffic.Q:
Can the Laurent series definition be extended to rational functions as well?
I know that the Laurent series definition can be extended to $\mathbb{C}$-valued functions. Indeed, one takes a polynomial $p(z)$ and considers the Taylor series for $p$ around the origin, and then, by setting $z=t+\frac{i}{2\pi}$ one gets the Laurent series for a function $p$.
Now, my question is, can we define a Laurent series for a rational function $f(z) = \frac{p(z)}{q(z)}$, where $p,q\in \mathbb{C}[z]$?
A:
Yes, you can define a Laurent series for any rational function. Let’s see what the definition is.
(Up to now, let’s suppose that $f$ is defined and is analytic in a neighborhood of $0$ in $\mathbb{C}$. In the question, we only consider $f(z) = \frac{p(z)}{q(z)}$ where $p,q$ are polynomials and $q(0)
eq0$.
Let $p(z)$ and $q(z)$ be polynomials with complex coefficients, and let $\alpha\in \mathbb{C}\setminus\mathbb{Z}$, $|\alpha|=1$. Let’s denote by $c_{\alpha}$ the coefficient of
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