 Rankin–Selberg integral for holomorphic cusp forms

For two weight-$k$ level-one cusp forms, use the primitive real-analytic Eisenstein series and $d\mu =dxdy/y^2$. The Rankin–Selberg unfolding identity isInitially this follows by absolute convergence, unfolding to $0\le x<1$, and integrating the Fourier series in $x$. Exponential cusp decay and polynomial growth of the primitive real-analytic Eisenstein series make the integral holomorphic for $\mathrm{Re}w>1$, continuing the Dirichlet series to $\mathrm{Re}s>k$.