Formelsammlung Mathematik: Identitäten: Partialbruchzerlegungen

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1

$\frac{{\alpha }^n{z}^{m-1}}{{\alpha }^n-z^n}=\frac{1}{n}\sum _{k=0}^{n-1}\frac{(\alpha {\xi }^k{)}^m}{\alpha {\xi }^k-z}\xi =\exp \left(\frac{2\pi i}{n}\right)$

2

$\frac{1}{(x-\alpha {)}^{n+1}(x-\beta {)}^{m+1}}=\sum _{k=0}^n\frac{(-1{)}^k}{(\alpha -\beta {)}^{m+1+k}}\frac{(m+k)!}{m!k!}\frac{1}{(x-\alpha {)}^{n+1-k}}+\sum _{k=0}^m\frac{(-1{)}^k}{(\beta -\alpha {)}^{n+1+k}}\frac{(n+k)!}{n!k!}\frac{1}{(x-\beta {)}^{m+1-k}}$

3

$\frac{t^m}{\prod _{k=1}^n(k^2+t)}=2\cdot \sum _{k=1}^n\frac{(-1{)}^{m+k-1}{k}^{2m+2}}{(n+k)!(n-k)!}\cdot \frac{1}{k^2+t}m<n$

4

$\frac{t^m}{\prod _{k=0}^n((2k+1{)}^2+t)}=\sum _{k=0}^n\frac{(-1{)}^{m+k}(2k+1{)}^{2m+1}}{2^{2n}(n+k+1)!(n-k)!}\cdot \frac{1}{(2k+1{)}^2+t}m\le n$