In mathematics one often combines multiple indexes together to form a multi- index. Generalizing formulas of one variable to several variables, for example, of power series in one variable to multiple power series, so it is usually sensible for notational reasons to use the multi- index notation. Formally speaking, is a multi- index a tuple of natural numbers.
Conventions of the multi- index notation
In this section, each tuple are natural numbers. For the multi- index notation usually the following conventions are adopted:
Where and denotes a differential operator.
A multiple power series can be written in short as.
If and are so true and.
For holds, where is.
If and, it is true or.
For and or what can be written in short as.
If and are m- times continuously differentiable functions, it shall
This identity is called the Leibniz rule.
And are m- times continuously differentiable functions, then
For multiple power series.
Are power series of one variable, so holds, where is.
And all are components of magnitude, the following applies.
If and are so true.
If and so true.
Cauchy's integral formula
In several variables can be the Cauchy integral formula
Write short as
Should be said. Likewise, the estimate holds, where is.
Is an analytic function or a holomorphic map, so you can use this function in a Taylor series
Develop, with a multi- index.
For with and apply.
This generalizes the Abelian identity.
The latter is obtained in the case.