MathematicsLet S=z∈ℂ : z2+z¯=0. Then ∑z∈S (Re⁡(z)+lm⁡(z)) is equal to ______.

Let $S = \{z \in ℂ : z^{2} + \bar{z} = 0\} . Then \sum_{z \in S} (Re (z) + lm (z))$ is equal to ______.

Solution:

$S = \{z \in C : z^{2} + \bar{z} = 0\}$

Let z = x + iy

$\begin{array}{l} z^{2} = x^{2} - y^{2} + 2 ixy \\ \bar{z} = x - iy \\ z^{2} + \bar{z} = x^{2} - y^{2} + x + i (2 xy - y) = 0 \\ \Rightarrow x^{2} + x - y^{2} = 0 & 2 xy - y = 0 \\ y = 0 or x = \frac{1}{2} \end{array}$

If y = 0; x = 0, - 1

$If x = \frac{1}{2}; y = \frac{\sqrt{3}}{2}, \frac{- \sqrt{3}}{2}$

$S = \{(0, 0), (- 1, 0), (\frac{1}{2}, \frac{\sqrt{3}}{2}), (\frac{1}{2}, \frac{- \sqrt{3}}{2})\}$

$\sum_{z \in S} Re (z) + Im (z) = (0 - 1 + \frac{1}{2} + \frac{1}{2}) + 0 + 0 + \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} = 0$

Let $S = \{z \in ℂ : z^{2} + \bar{z} = 0\} . Then \sum_{z \in S} (Re (z) + lm (z))$ is equal to ______.

Solution:

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