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In mathematics , several indices are often combined into a single multi-index . From a formal point of view, a multi-index is a tuple of natural numbers .

If you generalize formulas from one variable to several variables, it usually makes sense to use multi-index notation for reasons of notation . An example would be to rewrite a power series with a variable to multiple power series . Multi-indices are often used in multidimensional analysis and theory of distributions .

Multi-index notation conventions

In this section are each tuple of natural numbers. The following conventions are usually agreed for the multi-index notation:

where and denote a differential operator .

Application examples

Power series

A multiple power series can be briefly written as .

Power function

Is and are , then and .

Geometric series

For true , where is.

Binomial theorem

Are and is , then applies or .

Multinomial theorem

For and is or what can be briefly written as .

Leibniz rule

If and are m-times continuously differentiable functions, then applies



This identity is called the Leibniz rule .

And if there are m times continuously differentiable functions, then it is


where is.

Cauchy product

The following applies to series of multiple powers .

If power series are a variable, then where is.

Exponential series

For true .

Binomial series

If and are all components of absolute value , then applies .

Vandermonde convolution

Is and are , so applies .

Is and so is true .

Cauchy's integral formula

Cauchy's integral formula can be divided into several variables

write briefly as


where should be. The estimate where is also applies .

Taylor series

If it is an analytic function or a holomorphic mapping , this function can be converted into a Taylor series

develop, where is a multi-index.

Hurwitz identity

For with and applies .

This generalizes the Abelian identity .

The latter is obtained in the case .


  • Otto Forster: Analysis. Volume 2: differential calculus in R n . Ordinary differential equations. 7th improved edition. Vieweg + Teubner, Wiesbaden 2006, ISBN 3-8348-0250-6 ( Vieweg study. Basic course in mathematics ).
  • Konrad Königsberger : Analysis. Volume 2. 3rd revised edition. Springer-Verlag, Berlin et al. 2000, ISBN 3-540-66902-7 .