Let $z=x+i?y$ , where $y>0$ . Choose a positive real number $t$ so that $t^2=x^2+y^2$ and pick positive numbers $u$ and $v$ satisfying $u^2=(t+x)/2 , v^2=(t-x)/2$ . Show that $x+i?y=(u+i?v)^2$ . Can you do a similar procedure of finding a square root if y is negative?
I am not exactly sure how to start this problem. Am I free to pick any t and start from there? Also, how can I show that $x+i?y=(u+i?v)^2$?
