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Here are the definitions I'm working with:

Def 1. $f$ is called complex-analytic in $D$ if it has a power series expansion $$f(z) = \sum_{k_1, \ldots, k_n =1}^\infty c_{k_1\ldots k_n} (z_1-\alpha_1)^k_1 \ldots (z_n - \alpha_n)^{k_n}$$ Def 2. For holomorphicity, we can use the usual notion of differentiability in $\mathbb R^{2n}$ and convert it to $\mathbb C^n$. It is equivalent to $$\frac{\partial f}{\partial \bar{z}_k} (z_0) = 0 \qquad \forall k.$$

It is a standard result for $n=1$ but what happens for higher dimensions? My motivation stems from the definition of complex manifolds. In some sources the transition maps are required to be complex analytic while in others the requirement is holomorphicity.

Encyclopedia of mathematics says: "In analogy with the case of the plane the holomorphy of a function $f$ at a point $\alpha\in\mathbb C^n$ is equivalent to its expandability in a multiple power series in a neighbourhood of this point".

The important question is if they are indeed equivalent or only on a certain domain. If the latter is the case, this would imply that the different definitions of manifolds would differ considerably, which is a problem.

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  • $\begingroup$ Wikipedia might clarify the relatiin between analytic and holomorphic. $\endgroup$ Aug 19, 2022 at 0:05
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    $\begingroup$ Since you're concerned about the possible relevance of the domain, you should probably correct your definition of "complex analytic in $D$" so that it actually refers to $D$. $\endgroup$ Aug 19, 2022 at 0:29

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"Analytic" and "holomorphic" are equivalent, and no it does not depend on the domain. (At leas when analytic means "complex analytic".)

Both are local definitions. A function is analytic near a point if in some neighborhood of that point it is given by a convergent power series. A funcion is holomorphic if it is differentiable and satisfies the Cauchy-Riemann equations.

Both of these definitions just need to be checked in a neighborhood of every point, so the definitions do not depend on the domain.

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