Complex Analysis¶
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A function \(f(z)\) is analytic in a domain \(D\) if \(f(z)\) is single valued and has a finite derivative \(f'(z)\) for every \(z \in D\).
In fact analyticity is defined as a function that can be represented as convergent power series.
This is equivalent to holystic function, which is defined as differentiable in any neighbour within a certain domain.
Indeed, having a convergent power series is a quite strong condition. Even though we know Taylor series can always be a good estimation up to an error term \(O(x^n)\), Taylor series is not always convergent, even if it's convergent, it might not convergent to the function itself.
Take \(f(x) = e^{\frac{1}{-x^2}}\) as an example, and we additionally let \(f(0) = 0\). Then it's easy to check \(f^{(n)}(0)\) are always \(0\). So Taylor series is always zero no matter how many terms are used, which means it's convergent but obviously not convergent to the function itself.