Equivalence of Definitions of Weierstrass E-Function

Theorem
Definitions of Weierstrass E-Function are equivalent.

Definition 1 implies Definition 2
By Definition 1:


 * $ \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 ) $

By Taylor's Theorem, where expansion is done around $ \mathbf w = \mathbf z $ and Lagrange form of remainder is used:


 * $ \displaystyle F \left ( { x, \mathbf y, \mathbf w } \right ) = F \left ( { x, \mathbf y, \mathbf z } \right ) + \frac{ \partial F \left ( { x, \mathbf y, \mathbf z } \right ) }{ \partial \mathbf y' } \left ( { \mathbf w - \mathbf z } \right ) + \frac{ 1 }{ 2 } \sum_{ i, j = 1 }^n \left ( { w_i - z_i  } \right ) \left ( { w_j - z_j  } \right ) \frac{ \partial^2 F \left ( { x, \mathbf y, \mathbf z + \theta \left ( { \mathbf z - \mathbf w } \right ) } \right ) }{ \partial y_i' y_j' }  $

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

Insertion of this expansion into the definition for Weierstrass E-Function leads to the desired result.