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