Definition:Weierstrass E-Function/Definition 1

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:


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

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


 * $\map E {x, \mathbf y, \mathbf z, \mathbf w} = \map F {x, \mathbf y, \mathbf w} - \map F {x, \mathbf y, \mathbf z} + \paren {\mathbf w - \mathbf z} F_{\mathbf y'} \paren {x, \mathbf y, \mathbf z}$

Also see

 * Equivalence of Definitions of Weierstrass E-Function