"If a topological manifold $M$ has even dimension $n=2m$, we can identify $\mathbb{R}^{2m}$ with $\mathbb{C}^m$ and require the transition maps to be complex-analytic; this determines a complex-analytic structure. A manifold endowed with this structure is called a complex manifold."
This is a definition I encountered in John Lee's Introduction to Smooth Manifolds book in p.15. Here, what does it mean by "complex-analytic"? I have to deal with functions from open sets $\mathbb{C}^m$ to open sets of $\mathbb{C}^m$. Thus, the phrase "complex-analytic" would not mean the usual "holomorphic". Could anyone please explain what "complex anayltic" mean?