Definition:Weierstrass E-Function/Definition 2

Definition
Let $\mathbf y,\mathbf z,\mathbf w$ be $n$-dimensional vectors.

Let $\mathbf y$ be such that $\map{\mathbf y} a=A$ and $\map{\mathbf y} b=B$.

Let $J$ be a functional such that:


 * $\displaystyle J\sqbrk{\mathbf y}=\int_a^b \map F {x,\mathbf y,\mathbf y'}\rd x$

Let $\theta\in\R:0<\theta<1$.

The following mapping is known as the Weierstrass E-Function of $J\sqbrk{\mathbf y}$:


 * $\displaystyle \map E {x,\mathbf y,\mathbf z,\mathbf w}=\frac 1 2\sum_{i,k\mathop=1}^n\paren{w_i-z_i}\paren{w_k-z_k}F_{y_i'y_k'} \paren{x,\mathbf y,\mathbf z+\theta\paren{\mathbf w-\mathbf z} }$

Also see

 * Equivalence of Definitions of Weierstrass E-Function