Consider a function $f(z,q)$ ($|q| < 1$) meromorphic (at least) in the region $|z| < 2$, with a periodicity $f(zq,q) = f(z,q)$. So if one replace $a = e^{2\pi i \mathfrak{a}}$, $f$ is an elliptic function with modulus $\tau$ given by $q = e^{2\pi i \tau}$.
Suppose $f(z,q)$ has a simple pole at $z = z_0$ (and therefore an infinite series of poles $z_0q^k$, and an additional pole to cancel the residue) leading to a non-isolated singularity at $z = 0$.
My question is: is there any relation between $\operatorname{Res}_{z = z_0}z^{-1}f(z,q)$ and the "$\operatorname{Res}_{z = 0}z^{-1}f(z,q)$ " defined by first expanding $f(z,q) = \sum_{n} f_n(z)q^n$ and
$$
\begin{align}
\operatorname{Res}_{z = 0} \frac{ 1 }{ z }f(z,q) \equiv \sum_n \Bigg[ \operatorname{Res}_{z = 0}\frac{ 1 }{ z }f_n(z) \Bigg]q^n \ ?
\end{align}
$$
In certain physical problem, we encounter such a situation where the two residues satisfy common differential equations, and seem to be related by modular $S$-transformation. But we have no idea why it is the case.
Best Answer
There no residues for non-isolated singularities. If $f$ is meromorphic on $\Bbb{C}^*$ and $f(z)=f(zq)$ with $|q|<1$ then for any $p,r\in (|q|,1),k\in \Bbb{Z}$, $$I=\int_{|s|=p} f(s)\frac{ds}{is}=\int_{|s|=r q^k} f(s)\frac{ds}{is}$$ This is telling the $z^{-1}$ coefficient of the Laurent exansion of $f$ on any annulus $a<|z|<A$ not containing a pole.
Let $F(u)=f(e^{2i\pi u})$ which is meromorphic $\Bbb{Z+\frac{\log q}{2i\pi} Z}$ periodic, then $I=\int_0^1 F(u) du$, in general there is no closed-form for it, take a look at Weierstrass_eta_function which is $I$ when $F=\wp$.
We can find $I$ from the negative Laurent coefficients $c_{j,l}$ at the poles $b_j$ plus one constant Laurent coefficient and some $\zeta(b_i-b_j;\Lambda)$ constants, this is because $F-\sum_j \sum_{l=1}^{e_j} c_{j,l}\zeta(z-b_j;\Lambda)$ is entire and doubly periodic thus constant.
When the $b_j \in \Bbb{Q}\Lambda$ there are some algebraic relations and $I$ is algebraic in the $c_{j,l}$ and the $g_2,g_3$ parameters of the lattice.
Feel free to clarify what is your $f$ and what you'd like to do with $I$.