Complex Analysis

1. add Prove that $\overline{z^4}=\overline{z}{}^4$

   We will begin by proving that $\overline{z_1{z}_2}=\overline{z_1}\overline{z_2}$ for any two complex numbers $z_1$ and $z_2$

   Let $z_1=x_1+i{y}_1$ and $z_2=x_2+i{y}_2$. Then $$\begin{array}{cc}\overline{z_1{z}_2}& =\overline{(x_1+i{y}_1)(x_2+i{y}_2)}\\ & =\overline{x_1{x}_2+i(x_1{y}_2+y_1{x}_2)-y_1{y}_2}\\ & =x_1{y}_2-i(x_1{y}_2+y_1{x}_2)-y_1{y}_2\\ & =(x_1-i{y}_1)(x_2-i{y}_2)\\ & =\overline{(x_1+i{y}_1)}\overline{(x_2+i{y}_2)}\\ & =\overline{z_1}\overline{z_2}\end{array}$$

   The rest of the proof follows fairly simply. I will try to be as explicit as possible. We can write $$\overline{z^4}=\overline{z^2{z}^2}=\overline{z^2}\overline{z^2}=\left(\overline{z}\overline{z}\right)\left(\overline{z}\overline{z}\right)=\overline{z}{}^4$$

   Q.E.D.