Section 13: Analyzing the Annular Laurent Integrals

$$ \newcommand{\bint}{\displaystyle{\int\hspace{-10.4pt}\Large\mathit{8}}} \newcommand{\res}{\displaystyle{\text{Res}}} $$

The reader is asked to review Section 10: Region of convergence of algebraic power series for background on this section.

In this section we study the Laurent integrals of annular Puiseux series: $$ \begin{equation} w_n(z)= \sum_{k=0}^{\infty} a_k (z^{1/n})^k+\sum_{k=1}^{\infty} \frac{b_k}{\left(z^{1/n}\right)^k} \\ a_k=\frac{1}{2n\pi i} \bint \frac{w_n(z)}{\left(z^{1/n}\right)^{k+n}} dz\\ b_k=\frac{1}{2n\pi i} \bint w_n(z)\left(z^{1/n}\right)^{k-n} dz \end{equation} $$

using the function in Section 10 $$ \begin{equation} \begin{aligned} f(z,w)&=(-z^2+z^3)\\ &+(-4 z+3 z^2)w\\ &+(-z^3-9 z^4)w^2\\ &+(-2+8 z+4 z^2-4 z^3)w^3\\ &+(6-8 z^2+7 z^3+8 z^4)w^4 \end{aligned} \label{equation:equation20} \end{equation} $$ In Section 10, we computed the branch continuation table for this function, and it is shown again in Figure 1.

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