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# Formelsammlung Mathematik: Reihenentwicklungen

### Exponentialreihe

${\displaystyle \exp z=\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}\qquad z\in \mathbb {C} }$

### Logarithmus

${\displaystyle \ln(1-z)=-\sum _{k=1}^{\infty }{\frac {z^{k}}{k}}\qquad |z|\leq 1\,,\,z\neq 1}$

### Winkelfunktionen

#### Sinus

${\displaystyle \sin z=\sum _{k=0}^{\infty }(-1)^{k}{\frac {z^{2k+1}}{(2k+1)!}}\qquad z\in \mathbb {C} }$

#### Kosinus

${\displaystyle \cos z=\sum _{k=0}^{\infty }(-1)^{k}{\frac {z^{2k}}{(2k)!}}\qquad z\in \mathbb {C} }$

#### Tangens

${\displaystyle \tan z=\sum _{k=1}^{\infty }(-1)^{k}\,{\frac {B_{2k}\,2^{2k}\,(1-2^{2k})}{(2k)!}}\,z^{2k-1}=\sum _{k=1}^{\infty }{\frac {2\cdot \lambda (2k)}{\left({\frac {\pi }{2}}\right)^{2k}}}\,z^{2k-1}\qquad |z|<{\frac {\pi }{2}}}$

#### Kotangens

${\displaystyle \cot z=\sum _{k=0}^{\infty }(-1)^{k}{\frac {B_{2k}\,(2^{2k}-1)}{(2k)!}}\,z^{2k-1}=\sum _{k=0}^{\infty }{\frac {-2\,\zeta (2k)}{\pi ^{2k}}}\,z^{2k-1}\qquad 0<|z|<\pi }$

#### Sekans

${\displaystyle \sec z=\sum _{k=0}^{\infty }|E_{k}|\,{\frac {z^{k}}{k!}}=\sum _{k=0}^{\infty }{\frac {2\cdot \beta (2k+1)}{\left({\frac {\pi }{2}}\right)^{2k+1}}}\,z^{2k}\qquad |z|<{\frac {\pi }{2}}}$

#### Kosekans

${\displaystyle \csc z=\sum _{k=0}^{\infty }(-1)^{k}\,{\frac {(2-2^{2k})\,B_{2k}}{(2k)!}}\,z^{2k-1}=\sum _{k=0}^{\infty }{\frac {2\cdot \eta (2k)}{\pi ^{2k}}}\,z^{2k-1}\qquad 0<|z|<\pi }$

### Hyperbelfunktionen

#### Sinus Hyperbolicus

${\displaystyle \sinh z=\sum _{k=0}^{\infty }{\frac {z^{2k+1}}{(2k+1)!}}\qquad z\in \mathbb {C} }$

#### Kosinus Hyperbolicus

${\displaystyle \cosh z=\sum _{k=0}^{\infty }{\frac {z^{2k}}{(2k)!}}\qquad z\in \mathbb {C} }$

#### Tangens Hyperbolicus

${\displaystyle \tanh z=\sum _{k=1}^{\infty }{\frac {B_{2k}\,2^{2k}\,(2^{2k}-1)}{(2k)!}}\,z^{2k-1}\qquad |z|<{\frac {\pi }{2}}}$

#### Kotangens Hyperbolicus

${\displaystyle \coth z=\sum _{k=0}^{\infty }{\frac {B_{2k}\,2^{2k}}{(2k)!}}\,z^{2k-1}\qquad 0<|z|<\pi }$

#### Sekans Hyperbolicus

${\displaystyle {\text{sech}}\,z=\sum _{k=0}^{\infty }E_{k}\,{\frac {z^{k}}{k!}}\qquad |z|<{\frac {\pi }{2}}}$

#### Kosekans Hyperboliucs

${\displaystyle {\text{csch}}\,z=\sum _{k=0}^{\infty }{\frac {(2-2^{2k})\,B_{2k}}{(2k)!}}\,z^{2k-1}\qquad 0<|z|<\pi }$

### Arkusfunktionen

#### Arkussinus

${\displaystyle \arcsin z=\sum _{k=0}^{\infty }(-1)^{k}{-{\frac {1}{2}} \choose k}{\frac {z^{2k+1}}{2k+1}}=\sum _{k=0}^{\infty }{\frac {1}{2^{2k}}}{2k \choose k}{\frac {z^{2k+1}}{2k+1}}\qquad |z|\leq 1}$

#### Ausdruck mit Arkussinus

${\displaystyle {\frac {\arcsin z}{\sqrt {1-z^{2}}}}=\sum _{k=1}^{\infty }{\frac {(2z)^{2k-1}}{k\,{2k \choose k}}}\qquad |z|<1}$

#### Potenzen des Arkussinus

${\displaystyle \arcsin ^{2}z={\frac {1}{2}}\sum _{k=1}^{\infty }{\frac {(2z)^{2k}}{k^{2}\,{2k \choose k}}}\qquad |z|\leq 1}$

${\displaystyle \arcsin ^{3}z=6\sum _{k=0}^{\infty }\left[\sum _{m=0}^{k-1}{\frac {1}{(2m+1)^{2}}}\right]\,{\frac {1}{2^{2k}}}\,{2k \choose k}\,{\frac {z^{2k+1}}{2k+1}}\qquad |z|<1}$

${\displaystyle \arcsin ^{4}z=6\sum _{k=0}^{\infty }\left[\sum _{m=1}^{k-1}{\frac {1}{(2m)^{2}}}\right]\,{\frac {(2z)^{2k}}{k^{2}\,{2k \choose k}}}\qquad |z|<1}$

#### Arkuskosinus

${\displaystyle \arccos z={\frac {\pi }{2}}-\sum _{k=0}^{\infty }(-1)^{k}{-{\frac {1}{2}} \choose k}{\frac {z^{2k+1}}{2k+1}}={\frac {\pi }{2}}-\sum _{k=0}^{\infty }{\frac {1}{2^{2k}}}{2k \choose k}{\frac {z^{2k+1}}{2k+1}}\qquad |z|\leq 1}$

#### Arkustangens

${\displaystyle \arctan z=\sum _{k=0}^{\infty }(-1)^{k}{\frac {z^{2k+1}}{2k+1}}\qquad |z|\leq 1,z\neq \pm i}$

### Areafunktionen

#### Areasinus Hyperbolicus

${\displaystyle {\text{arsinh}}\,z=\sum _{k=0}^{\infty }{-{\frac {1}{2}} \choose k}{\frac {z^{2k+1}}{2k+1}}\qquad |z|\leq 1}$

#### Potenzen des Areasinus Hypoerbolicus

${\displaystyle {\text{arsinh}}^{2}z={\frac {1}{2}}\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}\,(2z)^{2k}}{k^{2}\,{2k \choose k}}}\qquad |z|\leq 1}$

#### Areatangens Hyperbolicus

${\displaystyle {\text{artanh}}\,z=\sum _{k=0}^{\infty }{\frac {z^{2k+1}}{2k+1}}\qquad |z|\leq 1,z\neq \pm 1}$

### Spezielle Funktionen

#### Zeta-Funktion

${\displaystyle \zeta (z)={\frac {1}{z-1}}+\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k!}}\,\gamma _{k}\,(z-1)^{k}}$

#### Gamma-Funktion

${\displaystyle \Gamma (1+z)=\sum _{n=0}^{\infty }\;\;\sum _{k_{1}+2k_{2}+...+nk_{n}=n}\!\!{\frac {(-\gamma )^{k_{1}}}{k_{1}!}}\prod _{m=2}^{n}{\frac {\left((-1)^{m}\,{\frac {\zeta (m)}{m}}\right)^{k_{m}}}{k_{m}!}}\;\;z^{n}}$

#### Digamma-Funktion

${\displaystyle \psi (1+z)=-\gamma +\sum _{k=1}^{\infty }(-1)^{k+1}\,\zeta (k+1)\,z^{k}}$

#### Bessel-Funktionen

${\displaystyle J_{\nu }(z)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k!\,\Gamma (\nu +k+1)}}\left({\frac {z}{2}}\right)^{\nu +2k}}$

#### Lambert W-Funktion

${\displaystyle W(z)=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}\,n^{n-1}}{n!}}\,z^{n}\qquad |z|<{\frac {1}{e}}}$

#### [Reihe mit Bessel-Funktion]

${\displaystyle e^{{\frac {z}{2}}\left(t-{\frac {1}{t}}\right)}=\sum _{n\in \mathbb {Z} }J_{n}(z)\,t^{n}}$

### Ausdrücke mit Winkelfunktionen

#### cos(αx)/sin(απ)

${\displaystyle \pi \,{\frac {\cos \alpha x}{\sin \alpha \pi }}=\sum _{k\in \mathbb {Z} }{\frac {(-1)^{k}\,\cos kx}{k+\alpha }}\qquad -\pi

#### sin(αx)/sin(απ)

${\displaystyle -\pi \,{\frac {\sin \alpha x}{\sin \alpha \pi }}=\sum _{k\in \mathbb {Z} }{\frac {(-1)^{k}\,\sin kx}{k+\alpha }}\qquad -\pi

#### cot(πz)

${\displaystyle \pi \,\cot \pi z=\sum _{k=-\infty }^{\infty }{\frac {1}{k+z}}\qquad z\in \mathbb {C} \setminus \mathbb {Z} }$

#### csc(πz)

${\displaystyle \pi \,\csc \pi z=\sum _{k=-\infty }^{\infty }{\frac {(-1)^{k}}{k+z}}\qquad z\in \mathbb {C} \setminus \mathbb {Z} }$

#### tan(πz)

${\displaystyle \pi \,\tan \pi z=\sum _{k=-\infty }^{\infty }{\frac {1}{k+{\frac {1}{2}}-z}}\qquad z\in \mathbb {C} \setminus \left\{k+{\frac {1}{2}}\,:\,k\in \mathbb {Z} \right\}}$

#### sec(πz)

${\displaystyle \pi \,\sec \pi z=\sum _{k=-\infty }^{\infty }{\frac {(-1)^{k}}{k+{\frac {1}{2}}-z}}\qquad z\in \mathbb {C} \setminus \left\{k+{\frac {1}{2}}\,:\,k\in \mathbb {Z} \right\}}$

#### sin(α arcsin(z))

${\displaystyle {\frac {\sin(\alpha \arcsin z)}{\alpha }}=\sum _{n=0}^{\infty }\prod _{k=1}^{n}\left[(2k-1)^{2}-\alpha ^{2}\right]{\frac {z^{2n+1}}{(2n+1)!}}}$

### Ausdrücke mit Wurzeln

#### 9.1

${\displaystyle {\frac {1}{2}}\left[\left({\sqrt {1+z^{2}}}+z\right)^{2\alpha }+\left({\sqrt {1+z^{2}}}-z\right)^{2\alpha }\right]=\sum _{n=0}^{\infty }\alpha \,{\frac {(\alpha +n-1)!}{(\alpha -n)!}}\,{\frac {(2z)^{2n}}{(2n)!}}}$

#### 9.2

${\displaystyle {\frac {1}{2}}\left[\left({\sqrt {1+z^{2}}}+z\right)^{2\alpha +1}-\left({\sqrt {1+z^{2}}}-z\right)^{2\alpha +1}\right]=\sum _{n=0}^{\infty }{\frac {2\alpha +1}{2}}\,{\frac {(\alpha +n)!}{(\alpha -n)!}}\,{\frac {(2z)^{2n+1}}{(2n+1)!}}}$

#### 9.3

${\displaystyle \left({\sqrt {1+z^{2}}}+z\right)^{2\alpha }=\sum _{k=0}^{\infty }{\frac {\alpha \,\left(\alpha +{\frac {k}{2}}-1\right)!}{\left(\alpha -{\frac {k}{2}}\right)!}}\,{\frac {(2z)^{k}}{k!}}}$

### Ausdrücke mit Hyperbelfunktionen

#### 10.1

${\displaystyle {\frac {\pi }{z}}\,{\frac {e^{\pi z}}{\cosh \pi z-\cos \pi z}}={\frac {1}{\pi z^{3}}}+{\frac {1}{z^{2}}}+{\frac {\pi }{2z}}+4\sum _{k=1}^{\infty }{\frac {1}{z^{2}+(z-2k)^{2}}}+8z\sum _{k=1}^{\infty }{\frac {k}{e^{2\pi k}-1}}\,{\frac {1}{z^{4}+4k^{4}}}}$

#### 10.2

${\displaystyle {\frac {\pi }{2}}\,{\frac {\sinh \pi z}{\cosh \pi z-\cos \pi z}}={\frac {1}{2z}}+\sum _{k=1}^{\infty }{\frac {z}{z^{2}+(z-2k)^{2}}}+\sum _{k=1}^{\infty }{\frac {z}{z^{2}+(z+2k)^{2}}}}$

### Rest

#### 11.1

${\displaystyle (1+z)^{1/z}=\sum _{n=0}^{\infty }\sum _{k=n}^{\infty }{\frac {1}{k!}}{\begin{bmatrix}k\\k-n\end{bmatrix}}\,z^{n}\qquad |z|\leq 1\;,\;z\neq 0,1}$

### Lagrange-Inversion

Zu ${\displaystyle x_{0},y_{0}\in \mathbb {C} }$ mit Umgebungen ${\displaystyle U(x_{0}),V(y_{0})\,}$ sei ${\displaystyle f:U(x_{0})\to V(y_{0})\;,\;x\mapsto y_{0}+\sum _{k=1}^{\infty }y_{k}(x-x_{0})^{k}}$ eine biholomorphe Funktion.
Für die Koeffizienten ${\displaystyle x_{n}\,(n\geq 1)}$ der Umkehrfunktion ${\displaystyle f^{-1}:V(y_{0})\to U(x_{0})\,,\,y\mapsto x_{0}+\sum _{k=1}^{\infty }x_{k}(y-y_{0})^{k}}$
gibt es die Formel ${\displaystyle x_{n}={\frac {1}{n!}}\,\left[{\frac {\mathrm {d} ^{n-1}}{\mathrm {d} x^{n-1}}}\left({\frac {x-x_{0}}{f(x)-f(x_{0})}}\right)^{n}\,\right]_{x\to x_{0}}}$.