Definition:Weierstrass E-Function/Definition 1

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:


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

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


 * $E \left({x, \mathbf y, \mathbf z, \mathbf w}\right) = F \left({x, \mathbf y, \mathbf w}\right) - F \left({x, \mathbf y, \mathbf z}\right) + \left({\mathbf w - \mathbf z}\right) F_{\mathbf y'} \left({x, \mathbf y, \mathbf z}\right)$

Also see

 * Equivalence of Definitions of Weierstrass E-Function