# 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:

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

where:

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

is the derivative of a vector-valued function.

### 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 $J\sqbrk{\mathbf y}$:

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

## Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.