identitate numere complexe via reziduri

[Cezar Lupu]

Fie \( z_{1}, z_{2}, \ldots, z_{n} \) numere complexe nenule distincte. Sa se arate ca pentru orice \( n\geq 3 \) are loc identitatea:

\( \sum_{k=1}^{n}\frac{1+z_{k}^{n-1}}{z_{k}^{2}}\prod_{j=1, j\neq k}^{n}\frac{1}{z_{k}-z_{j}}=\sum_{k=1}^{n}\frac{1}{z_{k}}\prod_{j=1}^{n}\frac{1}{z_{j}} \).