= Solution
For a compact subset $C\subset\mathfrak h$, the real-linear map
$$
\mathbb R^2\longrightarrow\mathbb C,
\qquad(c,d)\longmapsto c\tau+d
$$
has inverse norm bounded uniformly for $\tau\in C$. Hence there is $A_C>0$ such that
$$
|c\tau+d|\geq A_C\sqrt{c^2+d^2}.
$$
Therefore the absolute value of the defining series is bounded locally uniformly by
$$
A_C^{-k}\sum_{(c,d)\ne(0,0)}(c^2+d^2)^{-k/2},
$$
which converges for $k>2$. The <Weierstrass M-test> gives locally uniform absolute convergence, so termwise holomorphy proves that $G_k^{(x,y)}$ is holomorphic.
Back to article page
