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For a full Euclidean lattice $\Lambda \subseteq {\mathbb{R}}^n$, its dual lattice isFor a Schwartz function, the Poisson summation formula for a Euclidean lattice statesTo prove it, periodize $f$ over $\Lambda$. The resulting function on ${\mathbb{R}}^n/\Lambda$ has Fourier coefficient $m(\Lambda {)}^{-1}\hat{f}(\mu )$ at $\mu \in {\Lambda }^{\vee }$. Evaluating its absolutely convergent Fourier series at zero gives the identity. The same proof applies under the usual weaker hypotheses ensuring convergence of both sides.

Put $L_{\tau }=y^{-1/2}(\mathbb{Z}\tau +\mathbb{Z})$, a covolume-one lattice, and defineTermwise Mellin transformation in the initial half-plane gives

Split the integral at $t=1$. The integral over $[1,\infty )$ is entire in $s$ because the theta sum decays exponentially. Apply the Poisson summation formula for a Euclidean lattice and the Fourier eigenfunction calculation from part b to the interval $(0,1]$, then substitute $t\mapsto 1/t$. This rewrites the small-time integral as another exponentially convergent integral over $[1,\infty )$ plus explicit elementary Mellin terms. Those terms are meromorphic, but for positive even $k$ their apparent poles are canceled by the zeros of $1/\Gamma (s+k)$. The displayed formula therefore continues holomorphically to every $s\in \mathbb{C}$, proving the analytic continuation of a weight-k real-analytic Eisenstein series.