.. Example project, yo documentation master file, created by sphinx-quickstart on Tue Jan 31 18:33:14 2017. You can adapt this file completely to your liking, but it should at least contain the root `toctree` directive. .. _sect-theory_lattice_sums: Eisenstein-Rayleigh lattice sums ================================ Consider the lattice \ :math:`\mathcal{Q}` (see section [background]). For definiteness, it is assumed that :math:`\textrm{Im}\;\tau>0`, where :math:`\tau =\omega _{2}/\omega _{1}`. The Eisenstein summation is defined by the iterative sum .. math:: :label: LS20 \sum_{m_{1},m_{2}}=\lim_{N\rightarrow \infty }\sum_{m_{2}=-N}^{N}\left( \lim_{M\rightarrow \infty }\sum_{m_{1}=-M}^{M}\right) . \label{2.2} The lattice sums are introduced as follows .. math:: :label: LS21 S_{n}:=\sum_{m_{1},m_{2}}\;^{\prime }(m_{1}\omega _{1}+m_{2}\omega _{2})^{-n} \quad (n=2,3,\ldots), \label{2.1} where the prime means that :math:`m_{1}` and :math:`m_{2}` run over all integer numbers as in :eq:`LS20` except the pair :math:`(m_{1},m_{2})=(0,0)`. The sum :math:`S_{2}` is conditionally convergent and understood in the sense of the Eisenstein summation :eq:`LS20`. Though the rest sums :eq:`LS21` converge absolutely, the direct computations by :eq:`LS21` are problematic because of their slow convergence. The sum :math:`S_{2}` can be computed by a quick formula [#MIT2008BOOK]_ : .. math:: S_{2}= \left( \frac{\pi }{\omega _{1}}\right) ^{2}\left( \frac{1}{3}-8\sum_{m=1}^{\infty }\frac{mq^{2m}}{1-q^{2m}}\right) ,\text{ where }q=\exp \left( \pi i\tau \right). \label{2.3} It is known that :math:`S_{n}=0` for an odd :math:`n`. For an even :math:`n`, the sums :eq:`LS21` can be easily computed through the rapidly convergent infinite sums [#MIT2008BOOK]_ : .. math:: \begin{aligned} S_{4}=\frac{1}{60}\left( \frac{\pi }{\omega _{1}}\right) ^{4}\left( \frac{4}{3}+320\sum_{m=1}^{\infty }\frac{m^{3}q^{2m}}{1-q^{2m}}\right) ,\; \label{2.5} \\ S_{6}=\frac{1}{140} \left( \frac{\pi }{\omega _{1}}\right) ^{6}\left( \frac{8}{27}-\frac{448}{3}\sum_{m=1}^{\infty }\frac{m^{5}q^{2m}}{1-q^{2m}}\right).\end{aligned} The sums :math:`S_{2n}` (:math:`n\geq 4`) are calculated by the recurrence formula  [#MIT2008BOOK]_ : .. math:: S_{2n}=\frac{3}{\left( 2n+1\right) \left( 2n-1\right) \left( n-3\right) }\sum_{m=2}^{n-2}\left( 2m-1\right) \left( 2n-2m-1\right) S_{2m}S_{2(n-m)}. \label{2.7} The Eisenstein series are defined as follows [#WEIL1976]_ .. math:: :label: LS21 E_{n}(z):=\sum_{m_{1},m_{2}}(z-m_{1}\omega _{1}-m_{2}\omega _{2})^{-n}\,,\;n=2,3,...\;. \label{2.8} Each of the functions ([2.8]) is doubly periodic and has a pole of order :math:`n` at :math:`z=0`. Further, it is convenient to define the value of :math:`E_{n}(z)` at the origin as :math:`E_{n}(0):=S_{n}.` .. rubric:: References .. [#WEIL1976] A. Weil, *Elliptic Functions According to Eisenstein and Kronecker*, Springer-Verlag Berlin Heidelberg, 1976 .. [#MIT2008BOOK] V. Mityushev, E. Pesetskaya, S. V. Rogosin, *Analytical Methods for Heat Conduction in Composites and Porous Media*, in Cellular and Porous Materials: Thermal Properties Simulation and Prediction, A. Öchsner, G. E. Murch, M. J. S. de Lemos, eds., Wiley, 121-164, 2008.