How can I transform the fuction $x=y$ under the complex function $f(z)=z^2+z$???
First I define the horizontals and verticals lines of the complex plane. Then if $z=x+iy$ the function $z^2+z=x^2-y^2+2ixy+x+iy$. There is a horizontal line $l_1$, exists $c_1\in R$ such that $l_1$ is represented by $y=c_1$, for any $z\in l_1$ then $z=x+ic_1$. The next step is $z^2+z=x^2-c^2_1+2c_1ix+x+c_1i$ consequently $u=x^2+c^2_1+x$ and $v=2c_1x+c_1$. When $c_1$ its different to zero then $x=\frac{v-c_1}{2c_1}$ so we have $u=(\frac{v-c_1}{2c_1})^2-c^2_1$ for horizontal lines.
Analogously, if $l_2$ its a vertical line, then exists $c_2\in R$ such that $l_2$ is represented by $x=c_2$, for any $z\in l_2$ then $z=c_2+iy$. The next step is $z^2+z=c^2_2-y^2+2c_2iy+c_2+iy$ consequently $u=c^2_2-y^2+c_2$ and $v=2c_2y+y$ when $c_2$ its different to zero then $y=\dfrac{v}{2c_2+1}$ so we have $u=c^2_2-(\frac{v}{2c_2+1})^2+c_2$ for vertical lines.
Then the complex map is
Now if I have the function $x=y$. Then $z^2+z=x^2-x^2+2ix^2+x+ix=x+i(2x^2+x)$. I have a doubt about if my conversion is correct. Because it happens in intersections of my vertical lines with horizontal. How I can change the line to the complex plane?
