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let-put-w-e-i-2pi-n-calculate-S-n-k-0-n-1-1-x-w-k-and-W-n-k-0-n-1-1-x-w-k-2-

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Question Number 28432 by abdo imad last updated on 25/Jan/18
let put w=e^(i((2π)/n)) calculate S_n = Σ_(k=0) ^(n−1) (1/(x−w^k )) and W_n = Σ_(k=0) ^(n−1) (1/((x−w^k )^2 )) .
letputw=ei2πncalculateSn=∑k=0n−11x−wkandWn=∑k=0n−11(x−wk)2.
Commented by abdo imad last updated on 27/Jan/18
let introduce the polynomial p(x)= x^n −1 the roots of p(x)are the complex z_k =e^(i((2kπ)/n)) and k∈[[0,n−1]] we know that ((p^′ (x))/(p(x))) = Σ_(k=0) ^(n−1) (1/(x−z_k )) =Σ_(k=0) ^(n−1) (1/(x−w^k )) so S_n = ((nx^(n−1) )/(x^n −1)) also we have by derivation (d/dx)( ((p^′ (x))/(p(x))))=−Σ_(k=0) ^(n−1) (1/((x−w^k )^2 )) ⇒((p^(′′) (x)p(x) −(p^′ (x))^2 )/((p(x))^2 ))=−Σ_(k=0) ^(n−1) (1/((x−w^k )^2 )) W_n =−((n(n−1)x^(n−2) ( x^n −1) −(nx^(n−1) )^2 )/((x^n −1)^2 )) =−((n(n−1) x^(2n−2) −n(n−1)x^(n−2) −n^2 x^(2n−2) )/((x^n −1)^2 )) =−((−n x^(2n−2) −n(n−1)x^(n−2) )/((x^n −1)^2 ))=((n x^(2n−2) −n(n−1)x^(n−2) )/((x^n −1)^2 )) for n≥2 .
letintroducethepolynomialp(x)=xn−1therootsofp(x)arethecomplexzk=ei2kπnandk∈[[0,n−1]]weknowthatp′(x)p(x)=∑k=0n−11x−zk=∑k=0n−11x−wksoSn=nxn−1xn−1alsowehavebyderivationddx(p′(x)p(x))=−∑k=0n−11(x−wk)2⇒p″(x)p(x)−(p′(x))2(p(x))2=−∑k=0n−11(x−wk)2Wn=−n(n−1)xn−2(xn−1)−(nxn−1)2(xn−1)2=−n(n−1)x2n−2−n(n−1)xn−2−n2x2n−2(xn−1)2=−−nx2n−2−n(n−1)xn−2(xn−1)2=nx2n−2−n(n−1)xn−2(xn−1)2forn⩾2.

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