Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 137 / 1 / b / Solution

For a compact subset $C\subset \mathfrak{h}$, the real-linear maphas inverse norm bounded uniformly for $\tau \in C$. Hence there is $A_C>0$ such thatTherefore the absolute value of the defining series is bounded locally uniformly bywhich converges for $k>2$. The Weierstrass M-test gives locally uniform absolute convergence, so termwise holomorphy proves that $G_k^{(x,y)}$ is holomorphic.