Determine the reflection $z^{*}$ of $z$ across the parabola $y=x^2$. Expand $z^{*}$ into a power series in powers of $\bar{z}$...
I'm not looking for a complete solution, of course, because I genuinely don't know what the question is asking. This is Exercise 10.3.5 in Gamelin's "Complex Analysis" on the Schwarz Reflection Principle. Can someone give me a little bit of a push to start the question, and maybe some geometric intuition as to what's happening here? There's probably a purely algebraic way to do this but I want to get some intuition on this. I do not have an "attempt" as per the guidelines because I genuinely don't know where to start. If someone could give me a push in the right direction and maybe explain a little bit of the geometric ideas, I'd appreciate it.
The (algebraic) idea is to write the analytic curve (in this case the parabola $y=x^2$) as an equation in $z$ and $\bar{z}$, and solve it in the form $z=f(\bar{z})$ with an analytic function $f$. Since it is known that the analytic reflection is of the form $g(\bar{z})$ with an analytic function $g$, and since $f$ and $g$ agree on the complex conjugate of the curve, by the uniqueness principle you must have $f=g$, i.e., the original equation of the curve also gives you the reflection.
In the simple case of a circle $x^2+y^2=r^2$, it works as follows: Rewrite as $z\bar{z} = r^2$, solve for $z$ as $z = r^2/\bar{z}$, so the reflection is given by $z \mapsto r^2/\bar{z}$. In more complicated cases, solving the equation explicitly for $z$ might not be possible, but you should always be able to get as many derivatives at some point on the curve by implicit differentiation, or maybe some sort of series expansion.