Definition:Weierstrass E-Function/Definition 2

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Definition

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:

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

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

The following mapping is known as the Weierstrass E-Function of $J \sqbrk {\mathbf y}$:

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

Also see

• Equivalence of Definitions of Weierstrass E-Function

Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.

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