holomorphic function is real analytic?

Question

$f$ 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?

Answer 1

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.