I'm reading book by Motohashi: spectral theory of Riemann zeta function. And after defining the set of automorphic functions $L^2(\mathcal{F}, d{\mu})$ with Petersson inner product $$\langle f_1, f_2 \rangle = \int_{\mathcal{F}} f_1(z)\overline{f_2(z)}d\mu(z).$$
Then we claim that by Green's formula we have that $$\langle \Delta f_1, f_2 \rangle = \int_{\mathcal{F}} y^2 \nabla f_1(z) \cdot \overline{f_2(z)} d\mu (z).$$
We apply Green's formula to $$\langle \Delta f_1, f_2 \rangle = \lim_{Y \to \infty} \int_{\mathcal{F_y}} y^2 \nabla f_1(z)\overline{f_2(z)} d\mu (z).$$
$\mathcal{F}_y = \mathcal{F} \cap \{z: Im(z) \leq Y$ }.
$$= \lim_{Y \to \infty} \int_{\mathcal{F_y}} \nabla f_1(z) \cdot \overline{f_2(z)} dxdy - \lim_{Y \to \infty} \int_{\mathcal{\partial F_y}} y \frac{\partial f_1}{\partial y}(x+iY)\overline{f_2(x+iY)} dx.$$
There is more steps given in the book after this and those are enough for me to conclude.
But the question is: What is the Green's formula in this context and how do we apply it here?