Advanced Complex numbers

Question

Let $\omega$ be a nonreal root of $z^3 = 1.$ Let $a_1,$ $a_2,$ $\dots,$ $a_n$ be real numbers such that $$\frac{1}{a_1 + \omega} + \frac{1}{a_2 + \omega} + \dots + \frac{1}{a_n + \omega} = 2 + 5i.$$Compute $$\frac{2a_1 - 1}{a_1^2 - a_1 + 1} + \frac{2a_2 - 1}{a_2^2 - a_2 + 1} + \dots + \frac{2a_n - 1}{a_n^2 - a_n + 1}.$$

I have no clue how to do this. Can someone help?

Answer 1

Hint: $\overline \omega$ is another non-real root of $z^3=1$,

$$\sum_{j=1}^n\frac1{a_j+\overline\omega} = \overline{2+5i}$$

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$1,\omega, \overline\omega$ are roots of $z^3-1 = 0$, so

$$\omega\overline\omega = 1,\quad \omega+\overline\omega + 1 = 0$$

Consider the following sum,

$$\begin{align*} \sum_{j=1}^n\frac1{a_j+\omega} + \sum_{j=1}^n\frac1{a_j+\overline\omega} &= \sum_{j=1}^n\frac{(a_j+\omega)+(a_j+\overline\omega)}{(a_j+\omega)(a_j+\overline\omega)}\\ &= \sum_{j=1}^n\frac{2a_j+\omega+\overline\omega}{a_j^2 + \omega a_j +\overline\omega a_j + \omega\overline\omega}\\ &= \sum_{j=1}^n\frac{2a_j-1}{a_j^2 -a_j + 1}\\ \end{align*}$$

Answer 2

Rationalize the denominator for each term of the left hand side & the right hand side

For integer $n$ not divisible by$3$

Let $p=\dfrac{2n\pi}3,2\cos p=-1\ \ \ \ (1)$

$$\dfrac1{a_j+w}=\dfrac1{a_j+\cos p+i\sin p}=\dfrac{(a_j+\cos p)-i\sin p}{a_j^2+2a_j\cos p+1}$$

Use $(1)$

Equate the real parts