Question Number 148960 by tabata last updated on 01/Aug/21

find the resideo f(z)=(z/(z^n −1))

Answered by mathmax by abdo last updated on 02/Aug/21

les residus ici sont les poles de f et se sent les racines n^(eme) de lunite cad z_k =e^((i2kπ)/n) and k∈[[0,n−1]] ⇒f(z)=(z/(Π_(k=0) ^(n−1) (z−z_k ))) f(z)=Σ_(k=0) ^∞ (a_k /(z−z_k )) ⇒Res(f,z_k )=a_k =(z_k /(nz_k ^(n−1) )) =(z_k ^2 /n) autre methode f(z)=(z/((z−z_0 )....(z−z_(k−1) )(z−z_k )(z−z_(k+1) )...(z−z_(n−1) )) ⇒Res(f,z_k )=(z_k /((z_k −z_0 )...(z_k −z_(k−1) )(z_k −z_(k+1) )....(z_k −z_(n−1) )))