Complex power series property with real numbers

Question

I am trying to prove the following statement, seems clear to me but I am not able to give a formal proof. I have tried using the binomial expansion and Taylor's theorem for complex numbers.

Let $f(z)= \sum\limits_{n=0}^\infty c_nz^n$, $z \in \mathbb{C}$ be a power series with radius of convergence $R_1=\infty$.

Prove that $f(z-c), c \in \mathbb{R}$ can also be written as a power series $f(z-c)=\sum\limits_{n=0}^\infty b_nz^n$ with radius of convergence $R_2=\infty$.

Thanks in advance!

Answer 1

Hints

• Function $f(z)$ defined by the power series is analytic within the radius of converges
• Function $h(z)=z-c$ is analytic (entire in fact)
• $g(z)=(f\circ h)(z)$ is also analytic

You can tighten the result by the fact that $f(z)$ is in fact entire and deduct that $g(z)$ is also entire.