I'm reading [2] and got stuck in understanding the transition from the first to the second lines. I thought it was because of bivariate Taylor series expansion but couldn't quite reproduce this transition. Could someone explain it to me?
Notation
- K is a compact subset of $\mathbb{C}$
- $\omega \in \Omega$ where $\Omega$ is a fundamental parallelogram
- $g(w, z) = (1 - \frac{z}{\omega}) \exp (\frac{z}{\omega} + \frac{1}{2} (\frac{z}{\omega})^2)$ such that we were discussing to define the Weierstrass sigma-function; $\sigma(z) = z . \prod_{\omega \in \Omega - {0}} g(w, z)$.
Transition of interest (Page 93 of [2])
Reference
[2] Jones, Gareth A., and David Singerman. Complex functions: an algebraic and geometric viewpoint. Cambridge university press, 1987.