Multiindex

Potenzreihe

Eine Mehrfachpotenzreihe $\sum _{k_{1}\geq 0}\cdots \sum _{k_{n}\geq 0}a_{k_{1},\ldots ,k_{n}}(z_{1}-z_{1}^{o})^{k_{1}}\cdots (z_{n}-z_{n}^{o})^{k_{n}}$ lässt sich kurz schreiben als $\sum _{{\boldsymbol {k}}\geq 0}a_{\boldsymbol {k}}({\boldsymbol {z}}-{\boldsymbol {z}}^{o})^{\boldsymbol {k}}$ .

Potenzfunktion

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

Geometrische Reihe

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

Binomischer Lehrsatz

Sind ${\boldsymbol {x}},{\boldsymbol {y}}\in \mathbb {C} ^{n}$ und ist ${\boldsymbol {m}}\in \mathbb {N} ^{n}$ , so gilt $({\boldsymbol {x}}+{\boldsymbol {y}})^{\boldsymbol {m}}=\sum _{{\boldsymbol {k}}\leq {\boldsymbol {m}}}{{\boldsymbol {m}} \choose {\boldsymbol {k}}}{\boldsymbol {x}}^{\boldsymbol {k}}{\boldsymbol {y}}^{{\boldsymbol {m}}-{\boldsymbol {k}}}$ bzw. ${\frac {({\boldsymbol {x}}+{\boldsymbol {y}})^{\boldsymbol {m}}}{{\boldsymbol {m}}!}}=\sum _{{\boldsymbol {k}}+{\boldsymbol {j}}={\boldsymbol {m}}}{\frac {{\boldsymbol {x}}^{\boldsymbol {k}}}{{\boldsymbol {k}}!}}{\frac {{\boldsymbol {y}}^{\boldsymbol {j}}}{{\boldsymbol {j}}!}}$ .

Multinomialtheorem

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

Leibniz-Regel

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

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

beziehungsweise

{\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 $f_{1},\ldots ,f_{n}\colon \mathbb {R} \to \mathbb {R}$ m-mal stetig differenzierbare Funktionen, so ist

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

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

Cauchy-Produkt

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

Sind $f_{1}(z)=\sum _{\ell =0}^{\infty }a_{1\ell }z^{\ell }\;,\;\ldots \;,\;f_{n}(z)=\sum _{\ell =0}^{\infty }a_{n\ell }z^{\ell }$ Potenzreihen einer Veränderlichen, so gilt $f_{1}(z)\cdots f_{n}(z)=\sum _{\ell =0}^{\infty }\left(\sum _{|{\boldsymbol {k}}|=\ell }a_{\boldsymbol {k}}\right)z^{\ell }$ , wobei $a_{\boldsymbol {k}}=a_{1k_{1}}\cdots a_{nk_{n}}$ ist.

Exponentialreihe

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

Binomische Reihe

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

Vandermondesche Konvolution

Ist ${\boldsymbol {m}}\in \mathbb {N} ^{n}$ und sind ${\boldsymbol {\alpha }},{\boldsymbol {\beta }}\in \mathbb {C} ^{n}$ , so gilt ${{\boldsymbol {\alpha }}+{\boldsymbol {\beta }} \choose {\boldsymbol {m}}}=\sum _{{\boldsymbol {k}}\leq {\boldsymbol {m}}}{{\boldsymbol {\alpha }} \choose {\boldsymbol {k}}}{{\boldsymbol {\beta }} \choose {\boldsymbol {m}}-{\boldsymbol {k}}}=\sum _{{\boldsymbol {k}}+{\boldsymbol {j}}={\boldsymbol {m}}}{{\boldsymbol {\alpha }} \choose {\boldsymbol {k}}}{{\boldsymbol {\beta }} \choose {\boldsymbol {j}}}$ .

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

Cauchysche Integralformel

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

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

Taylor-Reihe

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

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.

Hurwitz-Identität

Für $x,y\in \mathbb {C}$ mit $x\neq 0$ und ${\boldsymbol {a}}=(a_{1},...,a_{n})\in \mathbb {C} ^{n}$ gilt $(x+y)^{n}=\sum _{{\boldsymbol {0}}\leq {\boldsymbol {k}}\leq {\boldsymbol {1}}}x\,(x+{\boldsymbol {a}}\cdot {\boldsymbol {k}})^{|{\boldsymbol {k}}|-1}\,(y-{\boldsymbol {a}}\cdot {\boldsymbol {k}})^{n-|{\boldsymbol {k}}|}$ .