Definition:Weierstrass E-Function

Definition 1
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 ) \mathrm d x $

Then the following mapping is known as Weierstrass E-Function:


 * $ \displaystyle E \left ( { x, \mathbf y, \mathbf z, \mathbf w } \right ) = F \left ( { x, \mathbf y, \mathbf w } \right ) - F \left ( { x, \mathbf y, \mathbf z } \right ) + \left ( { \mathbf w - \mathbf z } \right ) F_{ \mathbf y' } \left ( { x, \mathbf y, \mathbf z } \right ) $

Definition 2
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 ) \mathrm d x $

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

Then the following mapping is known as Weierstrass E-Function:


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