Answer to Question #118592 in Complex Analysis for Brute lakes

Question

im(z+9/z)=0

Answer 1

$Im(\frac{z+9}{z})= 0.......z = x+iy$

$Im (\frac{x+iy+9}{x+iy}) =0$

$Im(1+ \frac{9}{x+iy}) =0$

$Im(1+ \frac{9}{x+iy}\times \frac{x-iy} {x-iy}) =0$

$Im(1+ \frac{9\times (x-iy)}{x^2+y^2}) =0$

$Im(1+\frac {9x-9iy} {x^2+y^2}) =0$

$\frac{-9y}{x^2+y^2} = 0$

$y=0, z=x=Re(z)$

Answer :

$z=x=Re(z)$