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!


A convergent power series is always differrentiable, so you could do as you said.

In standard terminology, an analytic function is defined as a function with a convergent power series, while a holomorphic function is defined as one which is complex differrentiable. I.e., the definition starts both ways. An important result in complex analysis is that the two classes of functions exactly coincide, i.e., every analytic function is holomorphic and vice versa, so the two names get used interchangeably. Once you show this result, you can proceed the same way regardless of which definition you began with.

  • $\begingroup$ Hmmmm... I see, this is clarifying. $\endgroup$ – Macrophage May 24 '18 at 11:52
  • $\begingroup$ "I.e., the definition starts both ways." That's a strange way to put it. $\endgroup$ – zhw. May 24 '18 at 15:23
  • $\begingroup$ @zhw. It certainly is. But I think it gets the point across rhetorically. $\endgroup$ – BallBoy May 24 '18 at 15:26

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