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    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 &#8805; 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 &#8450;, and if &#8747;_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.