The next semester I'll be taking a math class where I'll learn about complex variable, so I'm starting to learn by myself the foundations. I have a couple of old homeworks from the past courses and I'm trying to answer them. I was attempting to do this question:
"Find and graph the region on the plane $w=u+iv$ delimited by the lines $x=1$, $y=1$ and $x+y=1$ using the transformation $w=-z^2$."
Well, the first thing I did was to determine what $u$ and $v$ are considering $z=x+iy$.
$$\begin{align*} w &= -z^2=-(x+iy)^2\\ &= -(x^2+2xyi-y^2)=\underbrace{y^2-x^2}_{u}+\underbrace{(-2xy)}_{v}i \end{align*}$$
Hence I have
$$ \left\lbrace \begin{array}{ccl} u(x,y) & = & y^2-x^2\\ v(x,y) & = & -2xy \end{array}\right.$$
Here I got a little confused. I have seen here other examples where it is given a certain value of $y$ and the recommendation is trying to eliminate either $x$ or $y$, but in this case, I'm not sure how to proceed. Any type of direction will be extremely helpful.