The locus of the complex number Z is a rectangle in the Argand diagram with corners $(-a,0), (a,0), (a,a)$, and $(-a,a)$, where $a>0$. What is the locus of $Z^2$?
It could be a relatively easy problem but I am struggling.
2026-04-26 00:21:49.1777162909
complex number locus
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1
Let's see. I'm pretty rusty at complex variables, but maybe I can get you started. It's easy to see what happens to the bottom side of the rectangle, on the $x$-axis; it folds at the origin, onto the interval $[0,a^2]$. Next I'd try to find the images of the other three sides of the rectangle. Let's start with the right side. Write it in parametric form: $x=a,y=at,0\le t\le1$. So $z^2=(a+iat)^2=a^2(1-t^2+2it)$ if I did the algebra right. Now I've got parametric equations for the image; eliminate the parameter . . . I think it's an arc of a parabola, but you'd better check my work. After you've got the image of the perimeter worked out, the locus you want is the region inside. The mapping $z\to z^2$ is one-to-one on the rectangle except for the part on the $x$-axis. To make the algebra easier, you might try to do it for $a=1$ first. The general case is going to look the same (I guess) except for scale.