Definition:Weierstrass E-Function

Definition
Let $\mathbf y, \mathbf z, \mathbf w : \R \to \R^n$ be $n$-dimensional vector-valued functions.

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

Let $F: \R^{2n + 1} \to \R$ be twice differentiable its (independent) variables.

Let $J$ be a functional such that:


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

where:


 * $\mathbf y' := \dfrac {\d \mathbf y} {\d x}$

is the derivative of a vector-valued function.

Also see

 * Equivalence of Definitions of Weierstrass E-Function