Section 13: Analyzing the Annular Laurent Integrals

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

This section is further study of the paper: On the Branching Geometry of Algebraic Functions

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: $$ \begin{equation} \begin{aligned} w_n(z)&= \sum_{k=0}^{\infty} a_k (z^{1/n})^k+\sum_{k=1}^{\infty} \frac{b_k}{\left(z^{1/n}\right)^k} \\ &=A(z)+S(z) \end{aligned} \end{equation} $$ with $$ \begin{equation} \begin{aligned} a_k&=\frac{1}{2n\pi i} \bint \frac{w_n(z)}{\left(z^{1/n}\right)^{k+n}} dz\\ b_k&=\frac{1}{2n\pi i} \bint w_n(z)\left(z^{1/n}\right)^{k-n} dz.\\ \label{equation:equation4} \end{aligned} \end{equation} $$ and $A(z)$ being the analytic terms of the series and $S(z)$, the singular terms. Or in symmetrical form: $$ \begin{equation} \begin{aligned} a_k&=\frac{1}{2n\pi i} \bint \frac{w_n(z)}{\left(z^{1/n}\right)^{k+n}} dz\\ w_n(z)&=\sum_{p=-\infty}^\infty a_p (z^{1/n})^p. \end{aligned} \label{equation:equation100} \end{equation} $$

We use 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. In this section, we analyze the integral expression of (\ref{equation:equation100}) in the following form: $$ \begin{equation} \begin{aligned} c_k&=\frac{1}{2n\pi i} \bint \frac{w_n(t)}{\left(z^{1/n}\right)^{k+n}} dz=\frac{1}{2 n \pi} \int_{t_0}^{t_e} \frac{w_n(t)re^{it}}{\left(re^{it}\right)^{\frac{k+n}{n}}}\\ &=\frac{1}{2 n\pi} \int_{t_0}^{t_e} w_n(t) \left(re^{it}\right)^{-k/n} dt\\ &=\frac{1}{2 n\pi r^{k/n}} \int_{t_0}^{t_e} w_n(t)\big[\cos(tk/n)-i\sin(tk/n)\big] dt\\ &=\frac{1}{2 n\pi r^{k/n}} I(k,n). \end{aligned} \label{eqn5} \end{equation} $$

We are however immediately confronted with oscillating integrals which increase in frequency as the parameter $k$ is increased. So the question arises: How do we compute (\ref{eqn5}) accurately?

Consider the analytic domains for the branches in the first annulus in the continuation table: $$ \begin{aligned} \text{Branch} & & \text{Domain} \\ \{1,3\} & & (0,0.00915) \\ \{2\} & & (0,0.693) \\ \{4\} & & (0,0.00915) \\ \end{aligned} $$

And we wish to study the power expansion for the $2$-cycle branch which we can calculate very precisely using Newton Polygon. How will those results compare with the numerical integration of (\ref{eqn5})? Or more importantly, how can we integrate (\ref{eqn5}) in order to achieve a desired accuracy with Newton Polygon as the benchmark?

First consider the integrand for the $\{1,3\}$ branch in annulus $1$. Figure 1 shows the real component of the integrand for various values of $k$ and illustrates how the integrand increases in frequency as $k$ increases.

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