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.
Re(z)=x,Im(z)=yRe(z)=x,Im(z)=yRe(z)=x,Im(z)=y
iz=−y+ixiz=-y+ixiz=−y+ix
Re(iz)=−y=Im(z)Re(iz)=-y=Im(z)Re(iz)=−y=Im(z)
Im(iz)=x=Re(z)Im(iz)=x=Re(z)Im(iz)=x=Re(z)
1/z=x−iy(x+iy)(x−iy)=x−iyx2+y21/z=\frac{x-iy}{(x+iy)(x-iy)}=\frac{x-iy}{x^2+y^2}1/z=(x+iy)(x−iy)x−iy=x2+y2x−iy
Re(1/z)=xx2+y2Re(1/z)=\frac{x}{x^2+y^2}Re(1/z)=x2+y2x
Im(1/z)=−yx2+y2Im(1/z)=-\frac{y}{x^2+y^2}Im(1/z)=−x2+y2y
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