The way I understand the principle is that we look at F, piecewise-defined as:
F= $$ z \mapsto f(z), \ z \in \Omega^+$$ $$ z \mapsto \overline{f(\bar z)}, \ z \in \Omega^-$$
Here $\Omega^+$ and $\Omega^-$ are symmetric with respect to the real axis.
Then F is the analytic continuation of f to the lower half space. With Morera's Theorem, F is shown to be analytic on the real axis, and so is analytic on the whole of $\Omega$.
Have I understood the Reflection Principle correctly, or am I making things up to convince myself that this is what the theorem is saying?
To justify that $\overline{f(\bar z)}$ is indeed analytic, I look at its power series, which is $\sum \bar a_n z^n$, but this is just the power series of the function f that was initially defined - and analytic - on the upper half space, except that the coefficients are now conjugated. Is this enough (even obvious?) to say that $\overline{f(\bar z)}$ is analytic, from looking at its power series? I am guessing it is, since the conjugation of z does not change the modulus of z, and all the convergence tests, e.g., ratio test, would give exactly the same convergence results that was gotten for the original function f.
Thanks,