In Chap 8 of Stein's complex analysis, he proved that all the conformal maps $f$ from the upper half plane $\mathbb{H}$ to a given polygon $P$ is of the form $c_1S(z)+c_2$, where $c_1, c_2$ are complex numbers and S(z) is the Schwarz-Christoffel integral.
In the proof, he proposed that for any vertice $a_k$ of the polygon, assign a new function$h_k(z)=(f(z)-a_k)^{1/\alpha_k}$, where $\alpha_k$ is the inner angle at $a_k$, then $h_k$ maps the segment $[A_{k-1},A_{K+1}]$ to a line segment$L_k$, and then he applied the Schwarch reflection principle to see that $h_k$ is analytically continuable to a holomorphic function in the two way infinite strip $A_{k-1}<Re(z)<A_{k+1}$, and then "since $h_k$ is injective up to $L_k$ the symmetry in the Schwarz reflection principle guarantees that $h_k$ is injective in the whole disc centered at $x$" and later on "finally the Schwarz reflection principle shows that $f$ is continuable in the exterior of a disk $|z|\leq R$, for large $R$"
Can anyone help explain how the Schwarz reflection principle is used in the two quoted text? Thank you very much!
Best Answer
So the usual reflection principle is for reflection across the real axis, assuming the function maps the real axis to the real axis. In general, you can generalize the Schwarz reflection principle to mappings from line segments or circular arcs to other line segments or circular arcs, the idea is just to find a linear fractional transformation that transfers your problem to that of the real axis to real axis mapping.
Thus, an analytic function that is defined on one side of a segment/circular arc which maps that segment/circular arc to another segment/circular arc can be extended across the segment/circular arc while preserving symmetries. (The symmetric point of a point inside a circle lies on the same radial line but goes from being distance $d$ from the center to being radius $R/d$ from the center where $R$ is the radius of the circle).
PS circle reflection problem: The Schwarz Reflection Principle for a circle , relevant to the second set of quotes perhaps.