Question #281345

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.


1
Expert's answer
2021-12-20T16:48:26-0500

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

iz=y+ixiz=-y+ix

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

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


1/z=xiy(x+iy)(xiy)=xiyx2+y21/z=\frac{x-iy}{(x+iy)(x-iy)}=\frac{x-iy}{x^2+y^2}


Re(1/z)=xx2+y2Re(1/z)=\frac{x}{x^2+y^2}


Im(1/z)=yx2+y2Im(1/z)=-\frac{y}{x^2+y^2}


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