Multiindex

In der Mathematik fasst man häufig mehrere Indizes zu einem einzigen Multiindex zusammen. Formal gesehen ist ein Multiindex α = ( α 1 , … , α n ) {\displaystyle {\boldsymbol {\alpha }}=(\alpha _{1},\ldots ,\alpha _{n})} ein Tupel natürlicher Zahlen.

Verallgemeinert man Formeln von einer Variable auf mehrere Variablen, so ist es aus notationstechnischen Gründen meist sinnvoll, die Multiindexschreibweise zu verwenden. Ein Beispiel wäre, eine Potenzreihe mit einer Veränderlichen auf Mehrfachpotenzreihen umzuschreiben. Multiindizes werden häufig in der mehrdimensionalen Analysis und Theorie der Distributionen verwendet.

In diesem Abschnitt seien α = ( α 1 , … , α n ) ,   k = ( k 1 , … , k n ) ,   ℓ = ( ℓ 1 , … , ℓ n ) ∈ N 0 n {\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}} jeweils n {\displaystyle n} -Tupel natürlicher Zahlen. Für die Multiindex-Schreibweise werden üblicherweise die folgenden Konventionen vereinbart:

k = ℓ ⟺ k 1 = ℓ 1 , … , k n = ℓ n k ≤ ℓ ⟺ k 1 ≤ ℓ 1 , … , k n ≤ ℓ n k + ℓ := ( k 1 + ℓ 1 , … , k n + ℓ n ) k ! := k 1 ! ⋯ k n ! ( α k ) := α ! ( α − k ) ! k ! = ( α 1 k 1 ) ⋯ ( α n k n ) | k | := k 1 + ⋯ + k n x k := x 1 k 1 ⋯ x n k n D k := D 1 k 1 ⋯ D n k 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}}}

wobei x ∈ C n {\displaystyle {\boldsymbol {x}}\in \mathbb {C} ^{n}} und D {\displaystyle {\boldsymbol {D}}} einen Differentialoperator bezeichnet.

Eine Mehrfachpotenzreihe ∑ k 1 ≥ 0 ⋯ ∑ k n ≥ 0 a k 1 , … , k n ( z 1 − z 1 o ) k 1 ⋯ ( z n − z n o ) k n {\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}}} lässt sich kurz schreiben als ∑ k ≥ 0 a k ( z − z o ) k {\displaystyle \sum _{{\boldsymbol {k}}\geq 0}a_{\boldsymbol {k}}({\boldsymbol {z}}-{\boldsymbol {z}}^{o})^{\boldsymbol {k}}} .

Ist x ∈ R n {\displaystyle {\boldsymbol {x}}\in \mathbb {R} ^{n}} und sind k , m ∈ N n {\displaystyle {\boldsymbol {k}},{\boldsymbol {m}}\in \mathbb {N} ^{n}} , so gilt D k x m m ! = x m − k ( m − k ) ! {\displaystyle {\boldsymbol {D}}^{\boldsymbol {k}}{\frac {{\boldsymbol {x}}^{\boldsymbol {m}}}{{\boldsymbol {m}}!}}={\frac {{\boldsymbol {x}}^{{\boldsymbol {m}}-{\boldsymbol {k}}}}{({\boldsymbol {m}}-{\boldsymbol {k}})!}}} und D k | x | m m ! = | x | m − | k | ( m − | k | ) ! {\displaystyle {\boldsymbol {D}}^{\boldsymbol {k}}{\frac {|{\boldsymbol {x}}|^{m}}{m!}}={\frac {|{\boldsymbol {x}}|^{m-|{\boldsymbol {k}}|}}{(m-|{\boldsymbol {k}}|)!}}} .

Für − 1 < x < 1 {\displaystyle -{\boldsymbol {1}}<{\boldsymbol {x}}<{\boldsymbol {1}}} gilt ∑ | k | ≥ 0 x k = 1 ( 1 − x ) 1 {\displaystyle \sum _{|{\boldsymbol {k}}|\geq 0}{\boldsymbol {x}}^{\boldsymbol {k}}={\frac {1}{({\boldsymbol {1}}-{\boldsymbol {x}})^{\boldsymbol {1}}}}} , wobei 1 = ( 1 , … , 1 ) {\displaystyle {\boldsymbol {1}}=(1,\ldots ,1)} ist.

Sind x , y ∈ C n {\displaystyle {\boldsymbol {x}},{\boldsymbol {y}}\in \mathbb {C} ^{n}} und ist m ∈ N n {\displaystyle {\boldsymbol {m}}\in \mathbb {N} ^{n}} , so gilt ( x + y ) m = ∑ k ≤ m ( m k ) x k y m − k {\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}}}} bzw. ( x + y ) m m ! = ∑ k + j = m x k k ! y j j ! {\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}}!}}} .

Für x = ( x 1 , … , x n ) ∈ R n {\displaystyle {\boldsymbol {x}}=(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}} und m ∈ N {\displaystyle m\in \mathbb {N} } ist ( x 1 + ⋯ + x n ) m = ∑ k 1 + ⋯ + k n = m ( m k 1 , … , k n ) x 1 k 1 ⋯ x n k 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}}} bzw. ( x 1 + ⋯ + x n ) m m ! = ∑ k 1 + ⋯ + k n = m x 1 k 1 k 1 ! ⋯ x n k 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}!}}} , was sich kurz schreiben lässt als | x | m m ! = ∑ | k | = m x k k ! {\displaystyle {\frac {|{\boldsymbol {x}}|^{m}}{m!}}=\sum _{|{\boldsymbol {k}}|=m}{\frac {{\boldsymbol {x}}^{\boldsymbol {k}}}{{\boldsymbol {k}}!}}} .

Ist m ∈ N n {\displaystyle {\boldsymbol {m}}\in \mathbb {N} ^{n}} und sind f , g : R n → R {\displaystyle f,g\colon \mathbb {R} ^{n}\to \mathbb {R} } m-mal stetig differenzierbare Funktionen, so gilt

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

beziehungsweise

( f g ) ( m ) m ! = ∑ k + j = m f ( k ) k ! g ( j ) j ! {\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}}!}}} .

Diese Identität heißt Leibniz-Regel.

Und sind f 1 , … , f n : R → R {\displaystyle f_{1},\ldots ,f_{n}\colon \mathbb {R} \to \mathbb {R} } m-mal stetig differenzierbare Funktionen, so ist

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

wobei f ( k ) = ( f 1 , … , f n ) ( ( k 1 ) , … , ( k n ) ) = f 1 ( k 1 ) ⋯ f n ( k n ) {\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})}} ist.

Für Mehrfachpotenzreihen f ( z ) = ∑ | ℓ | ≥ 0 a ℓ z ℓ , g ( z ) = ∑ | ℓ | ≥ 0 b ℓ z ℓ {\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 }}} gilt f ( z ) g ( z ) = ∑ | ℓ | ≥ 0 ( ∑ k + j = ℓ a k b j ) z ℓ {\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 }}} .

Sind f 1 ( z ) = ∑ ℓ = 0 ∞ a 1 ℓ z ℓ , … , f n ( z ) = ∑ ℓ = 0 ∞ a n ℓ z ℓ {\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 }} Potenzreihen einer Veränderlichen, so gilt f 1 ( z ) ⋯ f n ( z ) = ∑ ℓ = 0 ∞ ( ∑ | k | = ℓ a k ) z ℓ {\displaystyle f_{1}(z)\cdots f_{n}(z)=\sum _{\ell =0}^{\infty }\left(\sum _{|{\boldsymbol {k}}|=\ell }a_{\boldsymbol {k}}\right)z^{\ell }} , wobei a k = a 1 k 1 ⋯ a n k n {\displaystyle a_{\boldsymbol {k}}=a_{1k_{1}}\cdots a_{nk_{n}}} ist.

Für z = ( z 1 , . . . , z n ) ∈ C n {\displaystyle {\boldsymbol {z}}=(z_{1},...,z_{n})\in \mathbb {C} ^{n}} gilt e z 1 + . . . + z n = ∑ k ∈ N 0 n z k k ! {\displaystyle e^{z_{1}+...+z_{n}}=\sum _{{\boldsymbol {k}}\in \mathbb {N} _{0}^{n}}{\frac {{\boldsymbol {z}}^{\boldsymbol {k}}}{{\boldsymbol {k}}!}}} .

Sind α , x ∈ C n {\displaystyle {\boldsymbol {\alpha }},{\boldsymbol {x}}\in \mathbb {C} ^{n}} und sind alle Komponenten von x {\displaystyle {\boldsymbol {x}}} betragsmäßig < 1 {\displaystyle <1\,} , so gilt ( 1 + x ) α = ∑ | k | ≥ 0 ( α k ) x k {\displaystyle ({\boldsymbol {1}}+{\boldsymbol {x}})^{\boldsymbol {\alpha }}=\sum _{|{\boldsymbol {k}}|\geq 0}{{\boldsymbol {\alpha }} \choose {\boldsymbol {k}}}\,{\boldsymbol {x}}^{\boldsymbol {k}}} .

Ist m ∈ N n {\displaystyle {\boldsymbol {m}}\in \mathbb {N} ^{n}} und sind α , β ∈ C n {\displaystyle {\boldsymbol {\alpha }},{\boldsymbol {\beta }}\in \mathbb {C} ^{n}} , so gilt ( α + β m ) = ∑ k ≤ m ( α k ) ( β m − k ) = ∑ k + j = m ( α k ) ( β j ) {\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}}}} .

Ist m ∈ N {\displaystyle m\in \mathbb {N} } und α = ( α 1 , . . . , α n ) ∈ C n {\displaystyle {\boldsymbol {\alpha }}=(\alpha _{1},...,\alpha _{n})\in \mathbb {C} ^{n}} , so gilt ( | α | m ) = ∑ | k | = m ( α k ) {\displaystyle {|{\boldsymbol {\alpha }}| \choose m}=\sum _{|{\boldsymbol {k}}|=m}{{\boldsymbol {\alpha }} \choose {\boldsymbol {k}}}} .

In mehreren Veränderlichen z 1 , … , z n {\displaystyle z_{1},\ldots ,z_{n}\,} lässt sich die cauchysche Integralformel

D k f ( z 1 , … , z n ) k ! = 1 ( 2 π i ) n ∮ ∂ U n ⋯ ∮ ∂ U 1 f ( ξ 1 , … , ξ n ) ( ξ 1 − z 1 ) k 1 + 1 ⋯ ( ξ n − z n ) k n + 1 d ξ 1 ⋯ d ξ 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}}

kurz schreiben als

a k := D k f ( z ) k ! = 1 ( 2 π i ) 1 ∮ ∂ U f ( ξ ) ( ξ − z ) k + 1 d ξ {\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 }}} ,

wobei ∂ U = ∂ U 1 × ⋯ × ∂ U n {\displaystyle \partial {\boldsymbol {U}}=\partial U_{1}\times \cdots \times \partial U_{n}} sein soll. Ebenso gilt die Abschätzung | a k | ≤ M r k {\displaystyle |a_{\boldsymbol {k}}|\leq {\tfrac {M}{{\boldsymbol {r}}^{\boldsymbol {k}}}}} , wobei M = max ξ ∈ ∂ U | f ( k ) | {\displaystyle \textstyle M=\max _{{\boldsymbol {\xi }}\in \partial {\boldsymbol {U}}}|f({\boldsymbol {k}})|} ist.

Ist f : R n → R {\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} } eine analytische Funktion oder f : C n → C {\displaystyle f\colon \mathbb {C} ^{n}\to \mathbb {C} } eine holomorphe Abbildung, so kann man f {\displaystyle f} mit Hilfe eines Entwicklungspunktes z 0 ∈ R n {\displaystyle {\boldsymbol {z}}_{0}\in \mathbb {R} ^{n}} oder z 0 ∈ C n {\displaystyle {\boldsymbol {z}}_{0}\in \mathbb {C} ^{n}} in einer Taylorreihe

f ( z ) = ∑ k ∈ N 0 n D k f ( z 0 ) k ! ( z − z 0 ) k {\displaystyle f({\boldsymbol {z}})=\sum _{{\boldsymbol {k}}\in \mathbb {N} _{0}^{n}}{\frac {D^{\boldsymbol {k}}f({\boldsymbol {z}}_{0})}{{\boldsymbol {k}}!}}({\boldsymbol {z}}-{\boldsymbol {z}}_{0})^{\boldsymbol {k}}}

darstellen.

Für x , y ∈ C {\displaystyle x,y\in \mathbb {C} } mit x ≠ 0 {\displaystyle x\neq 0} und a = ( a 1 , . . . , a n ) ∈ C n {\displaystyle {\boldsymbol {a}}=(a_{1},...,a_{n})\in \mathbb {C} ^{n}} gilt ( x + y ) n = ∑ 0 ≤ k ≤ 1 x ( x + a ⋅ k ) | k | − 1 ( y − a ⋅ k ) n − | k | {\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}}|}} .

Dies verallgemeinert die Abelsche Identität ( x + y ) n = ∑ k = 0 n ( n k ) x ( x + a k ) k − 1 ( y − a k ) n − k {\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}\,x\,(x+ak)^{k-1}\,(y-ak)^{n-k}} .

Letztere erhält man im Fall a = ( a , a , . . . , a ) {\displaystyle {\boldsymbol {a}}=(a,a,...,a)} .

• Otto Forster: Analysis. Band 2: Differentialrechnung im Rⁿ. Gewöhnliche Differentialgleichungen. 7. verbesserte Auflage. Vieweg + Teubner, Wiesbaden 2006, ISBN 3-8348-0250-6 (Vieweg Studium. Grundkurs Mathematik).
• Konrad Königsberger: Analysis. Band 2. 3. überarbeitete Auflage. Springer-Verlag, Berlin u. a. 2000, ISBN 3-540-66902-7.