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.. _sect-wp_En:

Weierstrass :math:`\wp` and Eisenstein functions
================================================

The Eisenstein functions
:math:`E_n(z)` (:math:`n=2,3,\ldots`) and the Weierstrass function
:math:`\wp(z)` are related by the identities 


.. math::
   :label: Ek_formulas

   \begin{array}{cc}
       E_{2}(z)=\wp(z)+S_2,\\ \\
       E_{n}(z)=\frac{(-1)^n}{(n-1)!}\frac{d^{n-2}}{dz^{n-2}}\wp(z), \quad n=3, 4\ldots    \\
       \end{array}
   \label{eq:EtoP1}

where :math:`z\neq 0` and :math:`S_2` is a constant. It follows from the
elliptic function theory [#AKHIEZER]_ and from [#MIT2008BOOK]_ that

.. math::
   :label: wp_wp_bis

   \wp''(z)=6\wp(z)^2-30 S_4.
   \label{eq:wpBis}

Here :math:`S_2` and :math:`S_4` stand for lattice sums defined in
Appendix [eisenFun]. Thus each function :math:`E_n(z)` is an algebraic
combination of :math:`\wp(z)` and :math:`\wp(z)'`. One can implement
both functions :math:`\wp(z)` and :math:`\wp(z)'` via approximation of
the series expansion (see [#AKHIEZER]_, Table X, p. 204). Dependencies  :eq:`Ek_formulas` and :eq:`wp_wp_bis` be implemented in any Computer Algebra System, in order to calculate the symbolic
representations of \ :math:`E_n(z)`.Hence, to determine matrices  for
Eisenstein functions combined in a given basic sum, it is sufficient to
perform a set of algebraic operations on the *base* matrices
:math:`C_\wp` and :math:`C_\wp'`. In addition, it will be convenient to
define the value of :math:`\wp` and :math:`\wp'` at the origin as zero.
For the definition of :math:`E_n(z)` at the origin, see :ref:`section on lattice sums <sect-theory_lattice_sums>`.


.. rubric:: References

.. [#AKHIEZER] N. I. Akhiezer, *Elements of the Theory of Elliptic Functions*, American Mathematical Society, 1990.

.. [#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.