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I'm trying to understand the main differences (if there are) between analytic and holomorphic functions.

I know that a function is holomorphic in a connected open set if it is differentiable in each point.

Also I know that a function is analytic if it can be written in terms of Taylor expansion in each neighborhood of $x_0$.

So which are the differences between these two classes of functions? Or which topic should I use to show they are the same thing?

Thank you!

John Doe
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James Arten
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  • I had the same question once, got an answer off Quora, which may be helpful here. (Source) For complex functions they are equivalent, but not by definition the same. For a function to be analytic in a point, it needs to have a powerseries expansion around that point. For a function to be holomorphic in a point, it needs to be complex differentiable in that point. You can prove that the two conditions are equivalent for functions on the complex plane. – John Doe Mar 24 '19 at 16:57
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    It is quite obvious analytic functions are holomorphic. And that holomorphic functions are analytic is a consequence of the Cauchy integral formula. – reuns Mar 24 '19 at 17:27

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