Definition:Weierstrass E-Function/Definition 2

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

Let $\mathbf y$ be such that $\mathbf y \left({a}\right) = A$ and $\mathbf y \left({b}\right) = B$.

Let $J$ be a functional such that:


 * $\displaystyle J \left[{\mathbf y}\right] = \int_a^b F \left({x, \mathbf y, \mathbf y'}\right) \rd x$

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

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


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

Also see

 * Equivalence of Definitions of Weierstrass E-Function