= Anisotropic theta functional equation
{title2=$\theta_\Lambda(A)=\operatorname{covol}(\Lambda)^{-1}(\det A)^{-1/2}\theta_{\Lambda^*}(A^{-1})$}
For a full <Euclidean lattice> and a positive definite real symmetric matrix $A$, the <Gaussian theta sum> obeys
$$
\theta_\Lambda(A)=\operatorname{covol}(\Lambda)^{-1}(\det A)^{-1/2}\theta_{\Lambda^*}(A^{-1}).
$$
Use the <Fourier transform> kernel $e^{-2\pi i x\cdot z}$, the identity $\widehat{e^{-\pi x^TAx}}(z)=(\det A)^{-1/2}e^{-\pi z^TA^{-1}z}$, and the <Poisson summation formula for a Euclidean lattice>. No integrality or self-duality of the <Euclidean lattice> is assumed.
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