Weierstrass $$\wp$$ and Eisenstein functions¶

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

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

where $$z\neq 0$$ and $$S_2$$ is a constant. It follows from the elliptic function theory  and from  that

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

Here $$S_2$$ and $$S_4$$ stand for lattice sums defined in Appendix [eisenFun]. Thus each function $$E_n(z)$$ is an algebraic combination of $$\wp(z)$$ and $$\wp(z)'$$. One can implement both functions $$\wp(z)$$ and $$\wp(z)'$$ via approximation of the series expansion (see , Table X, p. 204). Dependencies  (1) and (2) be implemented in any Computer Algebra System, in order to calculate the symbolic representations of $$E_n(z)$$.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 $$C_\wp$$ and $$C_\wp'$$. In addition, it will be convenient to define the value of $$\wp$$ and $$\wp'$$ at the origin as zero. For the definition of $$E_n(z)$$ at the origin, see section on lattice sums.

References

  (1, 2) Akhiezer, Elements of the Theory of Elliptic Functions, American Mathematical Society, 1990.
  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.