I am self studying analytic number theory from Tom M Apostol Modular functions and Dirichlet series in number theory and I couldn't think about an argument in proof related to Modular group.
I am adding image of proof highlighting the argument which I don't understand.
My doubt is in 7 th line of 2nd image ie
$\phi $ is bounded in $R_\Gamma$ and it has been proved that $\phi $ is invariant under $\Gamma$ , so how does Apostol deduces $\phi $ is bounded in H?
Can someone please give a hint.
Since $R_\Gamma$ is a fundamental region for $\Gamma$, we have that $H = \Gamma \cdot R_\Gamma$. Stated differently, for any point $z \in H$, there is an element $\gamma \in \Gamma$ such that $\gamma z = z' \in R_\Gamma$. This is the defining characteristic of a fundamental domain.
Thus to understand the size of $\phi(z)$, you can use that $\phi(z) = \phi(\gamma z) = \phi(z')$, where we are using that $\phi$ is invariant under $\Gamma$. And thus $\phi$ being bounded on $R_\Gamma$ shows that $\phi$ is bounded on $H$.