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I am to answer the following question:

Show that every function $f$ which is analytic in a symmetric region $\Omega$ can be written in the form $f_1 + if_2$ where $f_1, f_2$ are analytic in $\Omega$ and real on the real axis.

I want to use the reflection principle, but I understand applying that when we are trying to extend the region. I'm confused on how to proceed. Thanks in advance.

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1 Answer 1

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Hint: $$f_1(z) = \frac{f(z) + \overline{f(\overline{z})}}{2}$$

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