Answer to Question #130154 in Complex Analysis for Rebecca Omweri

Question

Show that

a)z+z*=2 Re z=2x

b)z-z*=2i Im z=2iy

c)z/z*={x^2-y^2/x^2+y^2}+i{2xy/x^2+y^2}

Answer 1

$z=x+iy\newline z^*=x-iy\newline a) z+z^*=x+iy+x-iy=2x\newline b) z-z^*=x+iy-(x-iy)=2iy\newline c) \dfrac{z}{z^*}=\dfrac{x+iy}{x-iy}=\dfrac{(x+iy)*(x-iy)}{(x-iy)*(x-iy)}=\dfrac{x^2+2ixy+y^2}{x^2-y^2}=\dfrac{x^2+y^2}{x^2-y^2}+\dfrac{2ixy}{x^2-y^2}$