Answer to Question #281345 in Complex Analysis for auni

Question

Show that Re (iz) = −Im (z) and Im (iz) = Re (z). Evaluate Re (1/z) and Im (1/z) if

z = x + iy and z 6= 0.

Answer 1

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

$iz=-y+ix$

$Re(iz)=-y=Im(z)$

$Im(iz)=x=Re(z)$

$1/z=\frac{x-iy}{(x+iy)(x-iy)}=\frac{x-iy}{x^2+y^2}$

$Re(1/z)=\frac{x}{x^2+y^2}$

$Im(1/z)=-\frac{y}{x^2+y^2}$