# Multi-index

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 . ${\ displaystyle {\ boldsymbol {\ alpha}} = (\ alpha _ {1}, \ ldots, \ alpha _ {n})}$

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: ${\ displaystyle {\ boldsymbol {\ alpha}} = (\ alpha _ {1}, \ ldots, \ alpha _ {n}), \ {\ boldsymbol {k}} = (k_ {1}, \ ldots, k_ {n}), \ {\ boldsymbol {\ ell}} = (\ ell _ {1}, \ ldots, \ ell _ {n}) \ in \ mathbb {N} _ {0} ^ {n}}$${\ displaystyle n}$

${\ displaystyle {\ begin {array} {ccl} {\ varvec {k}} = {\ varvec {\ ell}} & \ iff & k_ {1} = \ ell _ {1} \;, \; \ ldots \ ;, \; k_ {n} = \ ell _ {n} \\\\ {\ varvec {k}} \ leq {\ varvec {\ ell}} & \ iff & k_ {1} \ leq \ ell _ {1 } \;, \; \ ldots \;, \; k_ {n} \ leq \ ell _ {n} \\\\ {\ boldsymbol {k}} + {\ boldsymbol {\ ell}} &: = & ( k_ {1} + \ ell _ {1} \;, \; \ ldots \;, \; k_ {n} + \ ell _ {n}) \\\\ {\ boldsymbol {k}}! &: = & k_ {1}! \ cdots k_ {n}! \\\\ {{\ varvec {\ alpha}} \ choose {\ varvec {k}}} &: = & {\ frac {{\ varvec {\ alpha} }!} {({\ varvec {\ alpha -k}})! \, {\ varvec {k}}!}} = {\ alpha _ {1} \ choose k_ {1}} \ cdots {\ alpha _ {n} \ choose k_ {n}} \\\\ | {\ boldsymbol {k}} | &: = & k_ {1} + \ cdots + k_ {n} \\\\ {\ boldsymbol {x}} ^ {\ boldsymbol {k}} &: = & x_ {1} ^ {k_ {1}} \ cdots x_ {n} ^ {k_ {n}} \\\\ {\ boldsymbol {D}} ^ {\ boldsymbol { k}} &: = & D_ {1} ^ {k_ {1}} \ cdots D_ {n} ^ {k_ {n}} \ ,, \ end {array}}}$

where and denote a differential operator . ${\ displaystyle {\ boldsymbol {x}} \ in \ mathbb {C} ^ {n}}$${\ displaystyle {\ boldsymbol {D}}}$

## Application examples

### Power series

A multiple power series can be briefly written as . ${\ displaystyle \ sum _ {k_ {1} \ geq 0} \ cdots \ sum _ {k_ {n} \ geq 0} a_ {k_ {1}, \ ldots, k_ {n}} (z_ {1} - z_ {1} ^ {o}) ^ {k_ {1}} \ cdots (z_ {n} -z_ {n} ^ {o}) ^ {k_ {n}}}$${\ displaystyle \ sum _ {{\ varvec {k}} \ geq 0} a _ {\ varvec {k}} ({\ varvec {z}} - {\ varvec {z}} ^ {o}) ^ {\ boldsymbol {k}}}$

### Power function

Is and are , then and . ${\ displaystyle {\ boldsymbol {x}} \ in \ mathbb {R} ^ {n}}$${\ displaystyle {\ boldsymbol {k}}, {\ boldsymbol {m}} \ in \ mathbb {N} ^ {n}}$${\ displaystyle {\ varvec {D}} ^ {\ varvec {k}} {\ frac {{\ varvec {x}} ^ {\ varvec {m}}} {{\ varvec {m}}!}} = {\ frac {{\ varvec {x}} ^ {{\ varvec {m}} - {\ varvec {k}}}} {({\ varvec {m}} - {\ varvec {k}})!} }}$${\ displaystyle {\ varvec {D}} ^ {\ varvec {k}} {\ frac {| {\ varvec {x}} | ^ {m}} {m!}} = {\ frac {| {\ varvec {x}} | ^ {m- | {\ varvec {k}} |}} {(m- | {\ varvec {k}} |)!}}}$

### Geometric series

For true , where is. ${\ displaystyle - {\ varvec {1}} <{\ varvec {x}} <{\ varvec {1}}}$${\ displaystyle \ sum _ {| {\ varvec {k}} | \ geq 0} {\ varvec {x}} ^ {\ varvec {k}} = {\ frac {1} {({\ varvec {1} } - {\ varvec {x}}) ^ {\ varvec {1}}}}}$${\ displaystyle {\ boldsymbol {1}} = (1, \ ldots, 1)}$

### Binomial theorem

Are and is , then applies or . ${\ displaystyle {\ boldsymbol {x}}, {\ boldsymbol {y}} \ in \ mathbb {C} ^ {n}}$${\ displaystyle {\ boldsymbol {m}} \ in \ mathbb {N} ^ {n}}$${\ displaystyle ({\ varvec {x}} + {\ varvec {y}}) ^ {\ varvec {m}} = \ sum _ {{\ varvec {k}} \ leq {\ varvec {m}}} {{\ varvec {m}} \ choose {\ varvec {k}}} {\ varvec {x}} ^ {\ varvec {k}} {\ varvec {y}} ^ {{\ varvec {m}} - {\ boldsymbol {k}}}}$${\ displaystyle {\ frac {({\ varvec {x}} + {\ varvec {y}}) ^ {\ varvec {m}}} {{\ varvec {m}}!}} = \ sum _ {{ \ varvec {k}} + {\ varvec {j}} = {\ varvec {m}}} {\ frac {{\ varvec {x}} ^ {\ varvec {k}}} {{\ varvec {k} }!}} {\ frac {{\ varvec {y}} ^ {\ varvec {j}}} {{\ varvec {j}}!}}}$

### Multinomial theorem

For and is or what can be briefly written as . ${\ displaystyle {\ boldsymbol {x}} = (x_ {1}, \ ldots, x_ {n}) \ in \ mathbb {R} ^ {n}}$${\ displaystyle m \ in \ mathbb {N}}$${\ displaystyle (x_ {1} + \ cdots + x_ {n}) ^ {m} = \ sum _ {k_ {1} + \ cdots + k_ {n} = m} {m \ choose k_ {1}, \ ldots, k_ {n}} x_ {1} ^ {k_ {1}} \ cdots x_ {n} ^ {k_ {n}}}$${\ displaystyle {\ frac {(x_ {1} + \ cdots + x_ {n}) ^ {m}} {m!}} = \ sum _ {k_ {1} + \ cdots + k_ {n} = m } {\ frac {x_ {1} ^ {k_ {1}}} {k_ {1}!}} \ cdots {\ frac {x_ {n} ^ {k_ {n}}} {k_ {n}!} }}$${\ displaystyle {\ frac {| {\ varvec {x}} | ^ {m}} {m!}} = \ sum _ {| {\ varvec {k}} | = m} {\ frac {{\ varvec {x}} ^ {\ boldsymbol {k}}} {{\ boldsymbol {k}}!}}}$

### Leibniz rule

If and are m-times continuously differentiable functions, then applies ${\ displaystyle {\ boldsymbol {m}} \ in \ mathbb {N} ^ {n}}$${\ displaystyle f, g \ colon \ mathbb {R} ^ {n} \ to \ mathbb {R}}$

${\ displaystyle (fg) ^ {({\ varvec {m}})} = \ sum _ {{\ varvec {k}} \ leq {\ varvec {m}}} {{\ varvec {m}} \ choose {\ varvec {k}}} f ^ {({\ varvec {k}})} g ^ {({\ varvec {m}} - {\ varvec {k}})}}$

respectively

${\ displaystyle {\ frac {(fg) ^ {({\ varvec {m}})}} {{\ varvec {m}}!}} = \ sum _ {{\ varvec {k}} + {\ varvec {j}} = {\ varvec {m}}} {\ frac {f ^ {({\ varvec {k}})}} {{\ varvec {k}}!}} {\ frac {g ^ {( {\ boldsymbol {j}})}} {{\ boldsymbol {j}}!}}}$.

This identity is called the Leibniz rule .

And if there are m times continuously differentiable functions, then it is ${\ displaystyle f_ {1}, \ ldots, f_ {n} \ colon \ mathbb {R} \ to \ mathbb {R}}$

${\ displaystyle {\ frac {(f_ {1} \ cdots f_ {n}) ^ {m}} {m!}} = \ sum _ {| {\ boldsymbol {k}} | = m} {\ frac { {\ varvec {f}} ^ {({\ varvec {k}})}} {{\ varvec {k}}!}}}$,

where is. ${\ displaystyle {\ boldsymbol {f}} ^ {({\ boldsymbol {k}})} = (f_ {1}, \ ldots, f_ {n}) ^ {{\ big (} (k_ {1}) , \ ldots, (k_ {n}) {\ big)}} = f_ {1} ^ {(k_ {1})} \ cdots f_ {n} ^ {(k_ {n})}}$

### Cauchy product

The following applies to series of multiple powers . ${\ displaystyle f ({\ varvec {z}}) = \ sum _ {| {\ varvec {\ ell}} | \ geq 0} a _ {\ varvec {\ ell}} \, {\ varvec {z}} ^ {\ varvec {\ ell}} \;, \; g ({\ varvec {z}}) = \ sum _ {| {\ varvec {\ ell}} | \ geq 0} b _ {\ varvec {\ ell }} \, {\ boldsymbol {z}} ^ {\ boldsymbol {\ ell}}}$${\ displaystyle f ({\ varvec {z}}) \, g ({\ varvec {z}}) = \ sum _ {| {\ varvec {\ ell}} | \ geq 0} \ left (\ sum _ {{\ varvec {k}} + {\ varvec {j}} = {\ varvec {\ ell}}} a _ {\ varvec {k}} \, b _ {\ varvec {j}} \ right) {\ varvec {z}} ^ {\ boldsymbol {\ ell}}}$

If power series are a variable, then where is. ${\ displaystyle f_ {1} (z) = \ sum _ {\ ell = 0} ^ {\ infty} a_ {1 \ ell} z ^ {\ ell} \;, \; \ ldots \;, \; f_ {n} (z) = \ sum _ {\ ell = 0} ^ {\ infty} a_ {n \ ell} z ^ {\ ell}}$${\ displaystyle f_ {1} (z) \ cdots f_ {n} (z) = \ sum _ {\ ell = 0} ^ {\ infty} \ left (\ sum _ {| {\ varvec {k}} | = \ ell} a _ {\ boldsymbol {k}} \ right) z ^ {\ ell}}$${\ displaystyle a _ {\ boldsymbol {k}} = a_ {1k_ {1}} \ cdots a_ {nk_ {n}}}$

### Exponential series

For true . ${\ displaystyle {\ boldsymbol {z}} = (z_ {1}, ..., z_ {n}) \ in \ mathbb {C} ^ {n}}$${\ displaystyle e ^ {z_ {1} + ... + z_ {n}} = \ sum _ {{\ varvec {k}} \ geq 0} {\ frac {{\ varvec {z}} ^ {\ boldsymbol {k}}} {{\ boldsymbol {k}}!}}}$

### Binomial series

If and are all components of absolute value , then applies . ${\ displaystyle {\ varvec {\ alpha}}, {\ varvec {x}} \ in \ mathbb {C} ^ {n}}$${\ displaystyle {\ boldsymbol {x}}}$${\ displaystyle <1 \,}$${\ displaystyle ({\ varvec {1}} + {\ varvec {x}}) ^ {\ varvec {\ alpha}} = \ sum _ {| {\ varvec {k}} | \ geq 0} {{\ boldsymbol {\ alpha}} \ choose {\ boldsymbol {k}}} \, {\ boldsymbol {x}} ^ {\ boldsymbol {k}}}$

### Vandermonde convolution

Is and are , so applies . ${\ displaystyle {\ boldsymbol {m}} \ in \ mathbb {N} ^ {n}}$${\ displaystyle {\ varvec {\ alpha}}, {\ varvec {\ beta}} \ in \ mathbb {C} ^ {n}}$${\ displaystyle {{\ varvec {\ alpha}} + {\ varvec {\ beta}} \ choose {\ varvec {m}}} = \ sum _ {{\ varvec {k}} \ leq {\ varvec {m }}} {{\ varvec {\ alpha}} \ choose {\ varvec {k}}} {{\ varvec {\ beta}} \ choose {\ varvec {m}} - {\ varvec {k}}} = \ sum _ {{\ varvec {k}} + {\ varvec {j}} = {\ varvec {m}}} {{\ varvec {\ alpha}} \ choose {\ varvec {k}}} {{\ boldsymbol {\ beta}} \ choose {\ boldsymbol {j}}}}$

Is and so is true . ${\ displaystyle m \ in \ mathbb {N}}$${\ displaystyle {\ boldsymbol {\ alpha}} = (\ alpha _ {1}, ..., \ alpha _ {n}) \ in \ mathbb {C} ^ {n}}$${\ displaystyle {| {\ varvec {\ alpha}} | \ choose m} = \ sum _ {| {\ varvec {k}} | = m} {{\ varvec {\ alpha}} \ choose {\ varvec {k}}}}$

### Cauchy's integral formula

Cauchy's integral formula can be divided into several variables${\ displaystyle z_ {1}, \ ldots, z_ {n} \,}$

${\ displaystyle {\ frac {D ^ {\ varvec {k}} f (z_ {1}, \ ldots, z_ {n})} {{\ varvec {k}}!}} = {\ frac {1} {(2 \ pi i) ^ {n}}} \ oint _ {\ partial U_ {n}} \ cdots \ oint _ {\ partial U_ {1}} {\ frac {f (\ xi _ {1}, \ ldots, \ xi _ {n})} {(\ xi _ {1} -z_ {1}) ^ {k_ {1} +1} \ cdots (\ xi _ {n} -z_ {n}) ^ {k_ {n} +1}}} d \ xi _ {1} \ cdots d \ xi _ {n}}$

write briefly as

${\ displaystyle a _ {\ varvec {k}}: = {\ frac {D ^ {\ varvec {k}} f ({\ varvec {z}})} {{\ varvec {k}}!}} = { \ frac {1} {(2 \ pi i) ^ {\ varvec {1}}}} \ oint _ {\ partial {\ varvec {U}}} {\ frac {f ({\ varvec {\ xi}} )} {({\ varvec {\ xi}} - {\ varvec {z}}) ^ {{\ varvec {k}} + {\ varvec {1}}}}} \, {\ varvec {d \ xi }}}$,

where should be. The estimate where is also applies . ${\ displaystyle \ partial {\ boldsymbol {U}} = \ partial U_ {1} \ times \ cdots \ times \ partial U_ {n}}$${\ displaystyle | a _ {\ varvec {k}} | \ leq {\ tfrac {M} {{\ varvec {r}} ^ {\ varvec {k}}}}}$${\ displaystyle \ textstyle M = \ max _ {{\ varvec {\ xi}} \ in \ partial {\ varvec {U}}} | f ({\ varvec {k}}) |}$

### Taylor series

If it is an analytic function or a holomorphic mapping , this function can be converted into a Taylor series${\ displaystyle f \ colon \ mathbb {R} ^ {n} \ to \ mathbb {R}}$${\ displaystyle f \ colon \ mathbb {C} ^ {n} \ to \ mathbb {C}}$

${\ displaystyle f ({\ varvec {z}}) = \ sum _ {| {\ varvec {k}} | \ geq 0} {\ frac {D ^ {\ varvec {k}} f ({\ varvec { z}} ^ {o})} {{\ varvec {k}}!}} (zz ^ {o}) ^ {\ varvec {k}}}$

develop, where is a multi-index. ${\ displaystyle {\ boldsymbol {k}}}$

### Hurwitz identity

For with and applies . ${\ displaystyle x, y \ in \ mathbb {C}}$${\ displaystyle x \ neq 0}$${\ displaystyle {\ boldsymbol {a}} = (a_ {1}, ..., a_ {n}) \ in \ mathbb {C} ^ {n}}$${\ displaystyle (x + y) ^ {n} = \ sum _ {{\ varvec {0}} \ leq {\ varvec {k}} \ leq {\ varvec {1}}} x \, (x + {\ boldsymbol {a}} \ cdot {\ boldsymbol {k}}) ^ {| {\ boldsymbol {k}} | -1} \, (y - {\ boldsymbol {a}} \ cdot {\ boldsymbol {k}} ) ^ {n- | {\ varvec {k}} |}}$

This generalizes the Abelian identity . ${\ displaystyle (x + y) ^ {n} = \ sum _ {k = 0} ^ {n} {n \ choose k} \, x \, (x + ak) ^ {k-1} \, ( y-ak) ^ {nk}}$

The latter is obtained in the case . ${\ displaystyle {\ boldsymbol {a}} = (a, a, ..., a)}$

## 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 .