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Let F be a holomorphic function defined in an open disk $D(x_0, r)$ where $x_0 ∈ R$. Define a function $f$ in the interval $(x_0 − r, x_0 + r)$ by $f(x) = F(x$) for all $x ∈ (x-0 − r, x_0 + r)$. Suppose that $f$ is a real-valued function on the interval $(x_0 − r, x_0 + r)$. Prove that f is analytic at $x_0$.

I know that since $F$ is holomorphic in the open disk, $F$ has the power series representation that is also analytic in the open disk.

How do I use that to prove the real-valued function is analytic at $x_0$?

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Let $f(z)= \sum_{n=0}^{\infty}a_n(z-x_0)^n$ be the power series representation in $D(x_0,r)$. Then

$f(x)= \sum_{n=0}^{\infty}a_n(z-x_0)^n$ for all $ x \in (x_0 − r, x_0 + r).$

Now use $f(x) \in \mathbb R$ for $ x \in (x_0 − r, x_0 + r)$ , to obtain that all $a_n$ are real.

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