The analytic extension of Eisenstein series is a basic tool for the spectral analysis of the Laplace-Beltrami operator \(\Delta_X\) on a locally symmetric space \(X\). In the case of Riemann surfaces of finite area but non compact (as \(\mathbb{H}^2/PSL_2(\mathbb{Z})\)) and more generally for locally symmetric spaces of \(\mathbb{Q}\)-rank one, \textit{A. Selberg} [Collected papers. Vol. 1. Springer-Verlag (1989; Zbl 0675.10001)] proved their meromorphy on the entire complex plane. \textit{L. D. Faddeev} [Tr. Mosk. Mat. O.-va 17, 323-350 (1967; Zbl 0201.41601)] and \textit{Y. Colin de Verdière} [C. R. Acad. Sci., Paris, Ser. I 293, 361-363 (1981; Zbl 0478.30035)] gave other proofs related to scattering theory. The author stresses here an approach based on the meromorphic extension of the resolvent function \(R_X(s)=(\Delta_X-s(1-s))^{-1}\) defined for Re \(s>1\). This \(L^2\)-resolvent function \(R_X(s)\) cannot be extended through the critical line Re \(s=1/2\) (which projects on the continuous spectrum in the spectral parameter \(\lambda=s(1-s)\)). However, by introducing the weighted spaces \(L^2_\delta(X)=\{f\in L^2_{\text{loc}}(X)\), \(e^{\delta d(x_0,\cdot)}f\in L^2(X)\},\) the author proves that the function \(R_X\) as a function with values in the bounded operators from \(L^2_\delta(X)\) into \(L^2_{-\delta}(X)\) admits an extension to the whole complex plane if \(\delta>0\). The meromorphic extension for \(R_X\) comes from Fredholm theory arguments applied to a parametrix constructed by gluing compact resolvents (related to the compact part of \(X\) and cuspidal part of \(L^2(X)\)) and resolvents associated to each cusp of \(X\): because of the geometry, the Laplacian \(\Delta_X\) acts there as the Laplacian on the real line, for which the analysis is quite easy. The meromorphy of Eisenstein series follows immediately from the meromorphy of the resolvent. Such meromorphic extension results between weighted spaces are quite common in Schrödinger scattering theory, e. g. \textit{A. Sa Baretto} and \textit{M. Zworski} [Commun. Math. Phys. 173, No. 2, 401-415 (1995; Zbl 0835.35099)], see also the Stanford Lectures \textit{Geometric scattering theory} by \textit{R. B. Melrose} [Cambridge Univ. Press (1995; Zbl 0849.58071)] and references therein for related works in global analysis.