Equivalence of Definitions of Weierstrass E-Function
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Theorem
Let $\mathbf y, \mathbf z, \mathbf w$ be $n$-dimensional vectors.
Let $\mathbf y$ be such that $\map{\mathbf y} a=A$ and $\map{\mathbf y} b=B$.
Let $J$ be a functional such that:
- $\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
The following definitions of the concept of Weierstrass E-Function are equivalent:
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 $J \sqbrk {\mathbf y}$:
- $\ds \map E {x, \mathbf y, \mathbf z, \mathbf w} = \frac 1 2 \sum_{i, k \mathop = 1}^n \paren {w_i - z_i} \paren {w_k - z_k} F_{y_i' y_k'} \paren {x, \mathbf y, \mathbf z + \theta \paren {\mathbf w - \mathbf z} }$
Proof
Definition 1 implies Definition 2
By Definition 1:
- $\map E {x, \mathbf y, \mathbf z, \mathbf w} = \map F {x, \mathbf y, \mathbf w} - \map F {x, \mathbf y, \mathbf z} + \paren {\mathbf w - \mathbf z} \map {F_{\mathbf y'} } {x, \mathbf y, \mathbf z}$
By Taylor's Theorem, where expansion is done around $\mathbf w = \mathbf z$ and Lagrange form of remainder is used:
- $\ds \map F {x, \mathbf y, \mathbf w} = \map F {x, \mathbf y, \mathbf z} + \frac {\partial \map F {x, \mathbf y, \mathbf z} } {\partial \mathbf y'} \paren {\mathbf w - \mathbf z} + \frac 1 2 \sum_{i, j \mathop = 1}^n \paren {w_i - z_i} \paren {w_j - z_j} \frac {\partial^2 \map F {x, \mathbf y, \mathbf z + \theta \paren {\mathbf z - \mathbf w} } } {\partial y_i'y_j'}$
where $\theta \in \R: 0 < \theta < 1$.
Insertion of this expansion into the definition for Weierstrass E-Function leads to the desired result.
$\Box$
Definition 2 implies Definition 1
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Sources
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 6.34$: The Weierstrass E-Function. Sufficient Conditions for a Strong Extremum