The mean values of the Weierstrass elliptic function ℘℘: Theory and application

• Derivation of differential equations satisfied by the mean values of ℘℘.
• Solution of these equations in terms of hypergeometric functions and Legendre functions.
• Numerical computation of the means for both real and complex valued invariants.
• Application of the results to vegetation patterning in semi-arid landscapes.

The Weierstrass elliptic functions can be parameterised using either lattice generators or invariants. Most presentations adopt the former approach. In this paper the authors give formulae that enable conversion between the two representations. Using these, they obtain differential equations satisfied by the mean values of ℘℘ over its periods; these mean values are considered as functions of the invariants. They show how to construct exact solutions for the means in terms of both hypergeometric functions and Legendre functions. These solutions are valid for both real and complex values of the invariants. For the case of real invariants, the authors prove various monotonicity results for the means with respect to the invariants. They also discuss the numerical computation of the means, and show a number of plots of the means against both real and complex valued invariants. Finally, they consider an application of their results to vegetation patterning in semi-arid landscapes.