For a region G⊂ℂ define G*:={z:z¯∈G} (where z¯ is the complex conjugate of z). If G is a symmetric region, that is G=G*, then we defineG+:={z∈G:Imz>0},G-:={z∈G:Imz<0} andG0:={z∈G:Imz=0}.
That is you can “reflect” an analytic function across the real axis. Note that by composing with various conformal mappings you could generalize the above to reflection across an analytic curve. So loosely stated, the theorem says that if an analytic function is defined in a region with some “nice” boundary and the function behaves “nice” on this boundary, then we can extend the function to a larger domain. Let us make this statement precise with the following generalization.
Theorem.
References
1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
2 John B. Conway. . Springer-Verlag, New York, New York, 1995.
