A version of the Schwartz reflection principle
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(A version of the Schwartz reflection principle.) Let the function f be continuous in the region { z : | z | < 1, Im z ≥ 0}, real valued on the segement ( - 1, 1) of the real axis, and holomorphic in the open set { z : | z | < 1, Im z > 0}. Use the result:
If f is a continuous complex-valued function in the open subset G of ℂ, and if ∫_R f(z)dz=0 for every rectangle R, with edges parallel to the coordinate axis, contained with its interior in G, then f is holomorphic;
to prove f can be extended holomorphically to the open unit disk.
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Solution Summary
A version of the Schwartz reflection principle is utilized.
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