I have figured out a nice way to prove that if the complex numbers [itex]z_1,z_2,\ldots, z_N\in\mathbb{C}[/itex] are all distinct, then the equation(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\prod_{n=1}^N \frac{1}{z - z_n} = \sum_{n=1}^N \frac{\alpha_n}{z-z_n}

[/tex]

is true for all [itex]z\in\mathbb{C}\setminus\{z_1,z_2,\ldots, z_N\}[/itex], where the alpha coefficients have been defined by the formula

[tex]

\alpha_n = \underset{n'\neq n}{\prod_{n'=1}^N} \frac{1}{z_n - z_{n'}}

[/tex]

I would like to leave the proof of this result as challenge to you guys, and I'm not in a need for advice myself at this point. Of course if somebody proves this in a way that is different from my proof, then I'm still reading.

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# I Partial Fraction Challenge

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