Hello

I am trying to find the asymptotic expansion for $t_1,t_2 \to 0+$ of the function

$$ F_{w_0,\tau}(t_1,t_2) = \sum_{w \in \mathbb{Z}\tau+\mathbb{Z}} w \ \operatorname{exp}(- |w|^2t_1 - |w_0-w|^2t_2) $$

where we may assume that $w_0 \in \mathbb{Z}\tau +\mathbb{Z}$. The obvious first step would probably be to try and use the transformation properties of the theta functions, but this seems to lead to a rather complicated expression involving rational functions in the $t_i$. Hope anyone has suggestions/references/etc.