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

Let $J$ be a functional such that:

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

where:

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

is the derivative of a vector-valued function.

### Definition 1

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

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

### Definition 2

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

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

$\ds \map E {x, \mathbf y, \mathbf z, \mathbf w} = \frac 1 2 \sum_{i, k \mathop = 1}^n \paren {w_i - z_i} \paren {w_k - z_k} F_{y_i' y_k'} \paren {x, \mathbf y, \mathbf z + \theta \paren {\mathbf w - \mathbf z} }$

## Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.