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
A multiple power series
can be briefly written as .
Is and are , then
and
.
For true
, where
is.
Are and is , then applies
or .
For and is
or what can be briefly written as .
Leibniz rule
If and are m-times continuously differentiable functions, then applies
respectively
-
.
This identity is called the Leibniz rule .
And if there are m times continuously differentiable functions, then it is
-
,
where is.
The following applies to series of multiple powers
.
If power series are a variable, then where is.
For true .
If and are all components of
absolute value , then applies .
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 .
literature
- 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 .