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I am an undergraduate student studying Complex Analysis.

Since holomorphic functions have properties that differentiable functions on a real line do not have in general (i.e. Taylor’s Thoerem, Open Mapping Property, etc.), I come into question about what property of complex plane results in this “nice” results on holomorphic functions.

In my textbook, written by Silverman, most of the properties come from the Taylor’s Thoerem. Thus my question might converge to, what difference between real line and complex plane makes every holomorphic function has their power series expansion?

In my first sight, I thought about the difference on an integral (on real line) and the ”line” integral (on complex plane), but this seems not very fundamental, since real multivariable functions also might not have the power series expansion.

I think the difference is related to their algebraic, or topological properties, but it is hard to reveal it for me.

It will be glad if someone give me an insight.

Thomas Andrews
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Alex Lee
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  • Fundamentally, there are multiple directions to approaching a limit in the complex numbers, which makes complex differentiation restrict the nature of the function much more. See this answer: https://math.stackexchange.com/a/3414254/7933 – Thomas Andrews Jun 03 '21 at 01:33
  • Here's another related post: https://math.stackexchange.com/questions/947235/why-does-being-holomorphic-imply-so-much-about-a-function – Joe Jun 03 '21 at 01:36
  • One major difference is that every non-zero polynomial has a root, since $\mathbb C$ is algebraically closed. This implies bounded entire functions are constant by Taylor's theorem. – Rushabh Mehta Jun 03 '21 at 01:49
  • This is a nice question . essentially comparing topologies of the complex plane and x,y plane – llecxe Jun 03 '21 at 18:59

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