Sketching Images of sets under a function in Complex Plane

Question

I want to know how I would be able to sketch the image of a function in the complex plane. Say I was considering the set A = {z = x + iy: x in (0,1) y in (0,1)} under the function f(z)=z^2. How would I sketch its image? The domain is clear and simple, but the issue with the image is I cannot work out how to sketch it, as I have, on an argand diagram, the real part in the x direction, and the imaginary part in the y direction, so where would the output value of f(z) be plotted?

Thanks in advance.

Answer 1

Note that $f(x+yi)=x^2-y^2+2xyi$. So, consider the line segments:

• from $0$ to $1$;
• from $1$ to $1+i$;
• from $1+i$ to $i$;
• from $i$ to $0$

and see whr $f$ maps them into. For instance, the second segment is $\bigl\{1+ti\,|\,t\in[0,1]\bigr\}$ and $f(1+ti)=1-t^2+2ti$. So, you get a parabolic arc.

Now, see what you get from the other three segmens. Doinf his will give you the boundary of $f(A)$.