# About the definition of holomorphic function

So I was aware that in my complex analysis textbook, holomorphic/analytic function is defined by complex differentiability in the neighborhood of any point in the domain. Then, in later chapters, the book introduces series.

But, as I'm doing a maths project elsewhere, I wonder if holomorphic functions can be defined first by having a complex power series converging to the function, and have the differentiability deduced from this definition, and every other theorems about analytic functions in complex analysis afterwards?

Hope somebody could guide me to understand the underlying logic behind those concepts, thanks!