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:


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

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