Multi-index power

Set

context	$G$ … group
context	$g \in \text{Sequence}(G)$
context	$\alpha \in \text{Sequence}(\mathbb N)$
context	$\mathrm{length}(g)=\mathrm{length}(\alpha)$

definiendum

$\langle g,\alpha\rangle \mapsto g^\alpha := \prod_{i=1}^{\mathrm{length}(\alpha)} g_i^{\alpha_i}$

We also write $|\gamma|=\sum_i^{\mathrm{length}(\gamma)} \gamma_i$ .

Discussion

In most cases, the base sequence is understood. E.g. if

$\gamma=\langle 3,1,0,0,2 \rangle$

is taken to be a multiindex, then $|\gamma|=6$ and we write

$f^{(\gamma)}(x) \equiv \frac{\partial^{|\gamma|}}{\partial x^\gamma} f \equiv \frac{\partial^3}{\partial x_1^3} \frac{\partial}{\partial x_2} \frac{\partial^2}{\partial x_5^2} f$

Reference

Wikipedia: Multi-index notation

Parents

Context

Group, Integer, Sequence length