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"A complex function is analytic at a point z if z is an interior point of some region where the function is analytic.". There is a requirement that the point be inside a region in which the function is analytic. Why isn't it enough for the function to be continuous in that region (except at that point)? Why does it have to be derivable? Moreover, what about the boundary points of that region? They may not be interior points of any other region, so how can they be analytic according to the definition above? One more question, I'm reading the Arfken mathematical methods for physicist and came to definition of an entire function: "If f(z) is analytic everywhere in the (finite) complex plane , we call it an entire function.". I don't understand the "finite" in the parenthesis. Why does the complex plane need to be finite here?

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    $\begingroup$ I can only comment on your second question. Forget about the "finite". There is also the extended complex plane where you add the point infinity to it (Riemannian sphere). So, "finite" here simply means $\mathbb C$. $\endgroup$ – Friedrich Philipp Mar 26 '16 at 7:13
  • $\begingroup$ I think restricting our definition of analyticity to interior points here is quite natural; The notion of differential at boundary points by requiring the existence of the limit of quotient is often not quite useful and often requiring an extention of the original function to larger domain which it is differentiable in the original sense is preferable. $\endgroup$ – cjackal Mar 26 '16 at 7:21
  • $\begingroup$ And I think the open condition for analyticity here is just the author's preference or convention; Usually one wants to develope the 1-1 correspondence between complex differentiability and analyticity(i.e, existence of local power series expansion), and only the point-differentiability does not guarantee such correspondence. $\endgroup$ – cjackal Mar 26 '16 at 7:25
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Definitions are made to give strength to the concepts or theorems they help in establishing. It turns out that continuity in the real line is not very different from continuity in the complex plane. What's interesting is that just by adding differentiability, you get the following without assuming anything else (you have to prove them, but they will follow from the properties of the complex plane):

1) The derivative so defined is a continuous function (Goursat's theorem)

2) There is a power series expansion of the function around every point in the domain,implying that the function is infinitely differentiable, and each derivative is continuous.

3) The function is zero if and only if there is one point at which all the derivatives are zero.

4)Liouville's theorem: All bounded entire functions are constants.

5)Little Picard' theorem: Every entire function is onto on $\mathbb{C}$ except for atmost one point.

6)Rouche's theorem, the fundamental proof assistant for irreducibility criteria of polynomials.

7) A quite unbelievable proof of the fundamental theorem of algebra in a few lines.

I'm not aware of what the finite in brackets means, but the strength that just differentiablitity gives you make it a must-want: Analytic functions are so much more "analytic" than continuous functions because of the few above properties coming by just adding differentiability in $\mathbb{C}$.

Analyticity applies only on open domains, because as you say, closed domains have boundary points which cannot be approached from certain directions, hence differentiability is not defined in such places. Furthermore, you can see that from the definition of power series: every power series converges in an open set (unless it is a constant), so it has got to be defined in a ball around some point for every point, so your domain has to be open. Differentiability in closed domains is dealt with using manifolds: That is a different subject altogether, and adds a whole new flavor to complex analysis.

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