I have this expression:
$$W(x,y)=\sum_{i=0}^{I}\sum_{j=0}^{J}W_{ij}P_i(x)P_j(y)$$
Where $I, J, W_{ij}$ are constants and $P_i , P_j$ are Legendre polynomials.
I want to compute the following equation:
$$F=\int_{-1}^1\int_{-1}^1\frac{W^2}{\sqrt{\frac{\partial^2W/\partial x^2 + \partial^2W/\partial y^2}{W}}}dxdy$$
I want to know if there is any relationship between the Legendre polynomial functions and their derivatives to further simplify this formulation to simply compute the integrals.