# Show that all coefficients of a complex power series with real only image vanish

Let $$R>0$$ and $$f: K_R(0) \rightarrow \mathbb{C}, f(z) = \sum_{k=0}^{\infty} a_kz^k$$ be a compley power series with convergence radius $$R>0$$ such that $$f(K_R(0)) \subset \mathbb{R}$$, where $$K_R(0)$$ is the disk around zero with radius R. I would like to show that all coefficients $$a_k$$ with $$k \in \mathbb{N}$$ vanish.

So far I have noticed that if the above statement is true, the power series must value the constant $$a_0$$ for any $$z \in K_R(0)$$ because otherwise the $$a_k$$ would not have to be zero for all $$k>0$$. My problem is that I do not see why there could not be any coefficients such that $$\sum_{k=1}^{\infty} a_kz^k$$ would be real as well.

My idea is to consider the coefficients in the form $$a_k = \frac{f^{(k)}(0)}{k}$$. If the power series $$f(z)$$ does value a constant $$a_0$$, then the derivatives $$f^{(k)}=0$$ , so that $$a_k = 0$$ for $$k \in \mathbb{N}$$.

Am I on the right path? Any hint would be highly appreciated.

A real valued analytic function on any domain is a constant. [One way to prove this is to use Cauchy-Riemann equations]. Since $$f$$ is anlytic it follows that $$f(z)=k_0$$ for all $$z$$.