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

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

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 . ${\displaystyle \sum _{k_{1}\geq 0}\cdots \sum _{k_{n}\geq 0}a_{k_{1},\ldots ,k_{n}}(z_{1}-z_{1}^{o})^{k_{1}}\cdots (z_{n}-z_{n}^{o})^{k_{n}}}$ ${\displaystyle \sum _{{\boldsymbol {k}}\geq 0}a_{\boldsymbol {k}}({\boldsymbol {z}}-{\boldsymbol {z}}^{o})^{\boldsymbol {k}}}$

Power function

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

Geometric series

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

Binomial theorem

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

Multinomial theorem

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

Leibniz rule

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

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

respectively

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

This identity is called the Leibniz rule .

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

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

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 . ${\displaystyle f({\boldsymbol {z}})=\sum _{|{\boldsymbol {\ell }}|\geq 0}a_{\boldsymbol {\ell }}\,{\boldsymbol {z}}^{\boldsymbol {\ell }}\;,\;g({\boldsymbol {z}})=\sum _{|{\boldsymbol {\ell }}|\geq 0}b_{\boldsymbol {\ell }}\,{\boldsymbol {z}}^{\boldsymbol {\ell }}}$ ${\displaystyle f({\boldsymbol {z}})\,g({\boldsymbol {z}})=\sum _{|{\boldsymbol {\ell }}|\geq 0}\left(\sum _{{\boldsymbol {k}}+{\boldsymbol {j}}={\boldsymbol {\ell }}}a_{\boldsymbol {k}}\,b_{\boldsymbol {j}}\right){\boldsymbol {z}}^{\boldsymbol {\ell }}}$

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

Exponential series

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

Binomial series

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

Vandermonde convolution

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

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

Cauchy's integral formula

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

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

write briefly as

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

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

Taylor series

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

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

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

Hurwitz identity

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

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

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 .