I'm reading about the $j$-function from Washington's book. We define $S_N$ to be the set of matrices of the form $\begin{bmatrix}a & b\\ 0 & d\end{bmatrix}$ where $a,b,d\in\mathbb{Z}, ad=N$ and $0 \le b < d$. Then, he says
For $S\in S_N$, the function $$j\circ S(\tau)=j\left(\frac{a \tau +b}{d}\right)$$ is analytic in $\mathfrak{h}$. Define $$F_N(X,\tau)=\prod_{S \in S_N}(X-(j\circ S)(\tau))=\sum_k a_k(\tau)X^k,$$ so $F_N$ is a polynomial in the variable $X$ with coefficients $a_k(\tau)$ that are analytic functions for $\tau \in \mathfrak{h}$.
I understand that $F$ would be an analytic function in $X$ because its a rational function and has a Laurent series expansion, whose coefficients would depend on $\tau$. I don't see why $a_k(\tau)$ would be analytic in $\tau$ themselves, though? I suppose this is something to do with Laurent series of several complex variables, that I haven't learnt about. Any explanation and reference would be appreciated!