Very simply, for what type of functions $f: \mathbb{C} \rightarrow \mathbb{C}$ is the following true? $f(\bar{z})=\overline{f(z)}$
Does Schwarz reflection principle imply this is true for all analytic functions?
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1Every $f: \mathbb{R} \to \mathbb{R}$ will do. – Yes Sep 18 '15 at 16:06
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Might I please see a proof (or link to one) ? – staple Sep 18 '15 at 16:07
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1Ah, just because the conjugate of a real number is again real. :) – Yes Sep 18 '15 at 16:08
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@staple You DO realize that $\bar\cdot |_{\mathbb R} \equiv \mathrm{id}$?? – AlexR Sep 18 '15 at 16:08
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And I chose to make it a comment instead of an answer because I want to hint you that you may have to put some additional conditions in order to assume away cases of less interest like the one I gave and hence your question can solicit more interesting answers. – Yes Sep 18 '15 at 16:09
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For one thing, ratios of polynomials with real coefficients. But there are many more. – André Nicolas Sep 18 '15 at 16:16
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How about among $f: \mathbb{C} \rightarrow \mathbb{C}$ – staple Sep 18 '15 at 16:23
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Sorry, did a bad job of qualifying my original question and reading Gudson's reply – staple Sep 18 '15 at 16:24