Let S= z ∈ C: z2+ barz=0 . Then displaystyle∑z=S( operatornameRe(z)+ operatornameIm(z)) is equal to

Question

Let S= z ∈ C: z2+ barz=0 . Then displaystyle∑z=S( operatornameRe(z)+ operatornameIm(z)) is equal to  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

S= z ∈ C: z2+ barz=0 Let z = x + iy z 2= x 2- y 2+2 ixy barz=x-i y z2+ barz=x2-y2+x+i(2 x y-y)=0 ⇒ x2+x-y2=0 2 x y-y=0 y=0 text or x=(1/2) If y =0 ; x =0,-1 If x=(1/2) ; y=(√3/2), (-√3/2) displaystyle∑z ∈ S( operatornameRe(z)+ operatornameIm(z)=(0-1+(1/2)+(1/2))+0+0+(√3/2)-(√3/2))

Q. Let S={z∈C:z2+zˉ=0}. Then z=S∑​(Re(z)+Im(z)) is equal to _______

S={z∈C:z2+zˉ=0} Let z=x+iy z2=x2−y2+2ixy zˉ=x−iy z2+zˉ=x2−y2+x+i(2xy−y)=0 ⇒x2+x−y2=0&2xy−y=0 y=0 or x=21​ If y=0;x=0,−1 If x=21​;y=23​​,2−3​​ z∈S∑​(Re(z)+Im(z)=(0−1+21​+21​)+0+0+23​​−23​​)

Q. Let $S = {z \in C : z^{2} + \overset{z}{ˉ} = 0}$ . Then $z = S \sum (Re (z) + Im (z))$ is equal to _______