Definition:Weierstrass E-Function

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Definition

Let $\mathbf y, \mathbf z, \mathbf w$ be $n$-dimensional vectors.

Let $\mathbf y$ be such that $\mathbf y \left({a}\right) = A$ and $\mathbf y \left({b}\right) = B$.

Let $J$ be a functional such that:

$\displaystyle J \left[{\mathbf y}\right] = \int_a^b F \left({x, \mathbf y, \mathbf y'}\right) \rd x$


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 $J\sqbrk{\mathbf y}$:

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


Also see


Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.