# 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 [1] and from [2] 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 [1], 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] (1, 2) Akhiezer, Elements of the Theory of Elliptic Functions, American Mathematical Society, 1990.
 [2] 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.