Complex Analytic Function's Power Series

Question

Studying complex function, I'm facing some fundamental question: Given open and bounded set $\Omega \in \mathbb{C}$ and $f:\Omega \rightarrow \Omega$ holomorphic on $\Omega$ which $f(z_{0})=z_{0}$. $f$ can be represented as power series around $z_{0}$ as $f(z)=\sum_{n=1}^{\infty}a_{n}(z-z_{0})^{n}$ so within the radius of convergence around $z_{0}$ the equality holds. therefore when I plug in $z_{0}$ I get $f(z_{0})= \sum _{n=1} ^{\infty}a_{n}(z_{0}-z_{0})^{n}=0$ so $z_{0}=0$. On the other hand, I can create the function $g(z)=z$ which is also holomorphic on $\Omega$ but for $0\neq z_{1}\in \Omega$ I also get that $z_{1}=0$ which contradicts with the definition of $f$. What am I missing?

Answer 1

You're missing the first term in the sum

$$ f(z) = \sum_{\color{red}{n = 0}}^{+\infty}a_n(z - z_0)^n $$

with this in mind

$$ f(z_0) = a_0 + a_1(z_0 - z_0) + \cdots = a_0 $$

In you case case $a_0 = z_0$