Multi-index notation

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.

Application Examples

Power series

A multiple power series can be written in short as.

Power function

If and are so true and.

Geometric series

For holds, where is.

Binomial theorem

If and, it is true or.

Multinomialtheorem

For and or what can be written in short as.

Leibniz rule

If and are m- times continuously differentiable functions, it shall

Respectively

This identity is called the Leibniz rule.

And are m- times continuously differentiable functions, then

Being.

Cauchy product

For multiple power series.

Are power series of one variable, so holds, where is.

Exponential

For true.

Binomial series

And all are components of magnitude, the following applies.

Vandermonde convolution

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.

Taylor series

Is an analytic function or a holomorphic map, so you can use this function in a Taylor series

Develop, with a multi- index.

Hurwitz identity

For with and apply.

This generalizes the Abelian identity.

The latter is obtained in the case.