### Volume 9, 2004. 36 - 43.

**Walter H. Chen and Michael S. Willis**

Properties of the Iterates of the Weierstrass-{\bf$\wp$} Function

**Abstract**. This paper discusses several properties of the Weierstrass-$\wp$ function, as defined on the fundamental parallelogram $\field{C}/\Gamma$, where $\field{C}$ is the complex plane and $\Gamma$ is the lattice generated by $\omega_1$ and $\omega_2$. Using the addition formula for $\wp(z_1 +z_2)$, we develop a reccurence relation for $\wp(nz)$ in terms of $\wp(z)$. We then examine the degree of this expression, some coefficients, and patterns concerning the poles of this function. We also consider the geometric interpretation of taking an arbitrary $z_0$ and adding it to itself, both in the fundamental parallelogram $\field{C}/\Gamma$ and the elliptic curve generated by $\wp(z)$ and $\wp'(z)$.

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Furman University Electronic Journal of Undergraduate Mathematics