Let S = z ∈ ℂ : z 2 + z ¯ = 0 . Then ∑ z ∈ S Re z + Im z is equal to _______.

Step-by-step solution

Given, S = z ∈ C : z 2 + z ¯ = 0 Let z = x + i y z 2 = x 2 - y 2 + 2 i x y z ¯ = x - i y Now putting the value in z 2 + z ¯ we get, z 2 + z ¯ = x 2 - y 2 + x + i 2 x y - y = 0 Now on comparing both side we get, ⇒ x 2 + x - y 2 = 0 and 2 x y - y = 0 ⇒ y = 0 or x = 1 2 Now if y = 0 ;   x = 0 , - 1 If x = 1 2   then  y = 3 2 , - 3 2 So, ∑ z ∈ S Re z + Im z = 0 - 1 + 1 2 + 1 2 + 0 + 0 + 3 2 - 3 2 = 0