The following is based on the topic Complex Mapping.
Example: Find the image of a straight line $2x + y = 1$ under the mapping $w = (1+i)z + 3i$.
Solution: Let us rewrite the mapping equation as $$ w = u+iv = (1+i)(x+iy) + 3+i \implies u+iv = x-y+3 + i(x+y+1), $$ which means that $$ u = x-y+3, \quad v = x+y+1. $$ Adding and subtracting these equations we obtain $$ u+v = 2x + 4, \quad v-u = 2y-2, $$ which leads to $$ x = \frac{1}{2}(u+v-4), \quad y = \frac{1}{2}(v-u+2). $$ Substituting these expressions into the equation of the straight line we obtain $$ u + v - 4 + \frac{1}{2}(v-u+2) = 1 \implies u + 3v = 8. $$
$\quad %ugly but working, and this formating seems relevant for the question$ Let us now consider a circle equation $|z-z_0| = R$. To obtain the equation of its image we substitute Eq. $(51)$ into this equation. As a result we have $$ \left| \frac{w-b}{a} - z_0 \right| = R. $$ Multiplying this equation by $|a|$ we obtain $$ |w - w_0| = R|a|, \quad w_0 = b + a z_0. $$ This is the equation of a circle centred at $w_0$ and a radius $R|a|$. hence the linear mapping maps a cirle into a circle.
However the previous questions were easy to follow, but this one seems to be hard to understand and researching online didn't help.
I sort of get the first bit with the sorting and then substituting, but then it reads "Let us now consider a circle of equations..." etc. And from here it goes to explain how this is used to find a solution, but its confusing and I feel like it's completely out of place. Can anyone - even if you don't understand complex mapping, figure out if this is a mistake by having a quick look of before and after this point to see if they flow right?
If its no problem, I would really appreciate how he gets from the first bit to the last bit. Just before the the circle comes up the final thing is "$\implies u + 3v = 8$" some clarification of how these final steps are found would also be helpful.
There is no other way to confirm if there is an error in the sheet so I'm hoping someone who understands this topic can help me figure this out.
The example is really two examples at once. The author considers one function, namely $f \colon \mathbb{C} \to \mathbb{C}$ with $f(z) = (1+i)z + 3+i$, and two geometricly interesting subsets of the complex plane, namely the line $$ L = \{x+iy \in \mathbb{C} \mid 2x+y = 1\} $$ and the circle $$ C = \{z \in \mathbb{C} \mid |z-z_0| = R\}. $$ It is then shown that the image $f(L)$ is also a line and the image $f(C)$ is also a circle. Aside from using the same function $f$ both examples are independent of each other.
The author could have done a better job by making clear that he will consider two examples; it should start with: