= Rankin–Selberg integral for holomorphic cusp forms
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{title2=$I(f,g,w)=\int_{\Gamma\backslash\mathbb H}f\overline g E(\tau,w)y^k\,d\mu$}
= Rankin–Selberg integral
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{synonym}
For two weight-$k$ level-one <cusp forms>, use the <primitive real-analytic Eisenstein series> and $d\mu=dx\,dy/y^2$. The <Rankin–Selberg unfolding> identity is
$$
I(f,g,w)=\frac{\Gamma(w+k-1)}{(4\pi)^{w+k-1}}\sum_{n\geq1}\frac{a_n\overline{b_n}}{n^{w+k-1}}.
$$
Initially this follows by absolute convergence, unfolding to $0\leq x<1$, 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 $\operatorname{Re}w>1$, continuing the <Dirichlet series> to $\operatorname{Re}s>k$.
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