If $u$ is real valued function on $\Omega \subset \Bbb R^n$ then what is meant by $D^{\alpha} u$?

Question

If $u$ is real valued function on $\Omega \subset \Bbb R^n$ such that $u \in C^{\infty} (\Omega)$ then what is meant by $D^{\alpha} u$ where $\alpha$ is a multi-index.I need this in order to understand Cauchy's estimate.

Answer 1

Let $\alpha := (\alpha_1,\alpha_2, \cdots , \alpha_n)$. Then

$$D^{\alpha} := {\frac {{\partial}^{\alpha_1}} {{\partial} {x_1}^{\alpha_1}}} {\frac {{\partial}^{\alpha_2 }} {{\partial} {x_2}^{\alpha_2}}} \cdots {\frac {{\partial}^{\alpha_n}} {{\partial} {x_n}^{\alpha_n}}} = \frac {{\partial}^{|\alpha|}} {{{\partial} {x_1}^{\alpha_1}}{{\partial} {x_2}^{\alpha_2}} \cdots {{\partial} {x_n}^ {\alpha_n}}},$$ where $|\alpha| = \alpha_1 + \alpha_2 + \cdots + \alpha_n$, $\alpha_i \in \Bbb N$ for $i= 1,2, \cdots , n$.