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$f$ is a holomorphic function on $\mathbb C^n$. If we regard $f$ as a function $F$ from $\mathbb R^{2n} \to \mathbb R^2$, is it necessarily that $F$ is real analytic?

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Of course, since holomorphic function has a power series expansion equal to itself in a proper neighborhood of any point in the domain, and you can write a complex power series to its real part and imaginary part and you'll get the real power series expansion. –  Yuchen Liu Apr 30 '12 at 9:24
    
@jerrysciencemath You can post your comment as an answer. –  Davide Giraudo May 1 '12 at 9:03
    
@DavideGiraudo Thanks! –  Yuchen Liu May 1 '12 at 9:48
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up vote 3 down vote accepted

Of course, since holomorphic function has a power series expansion equal to itself in a proper neighborhood of any point in the domain, and you can write a complex power series to its real part and imaginary part and you'll get the real power series expansion.

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