This question is concerned with the long-standing problem confusing so many people which is: how is it that we can view $z$ and $\bar{z}$ as independent variables. To be more precise, I understand all the formal manipulations using the chain rule and the trick with $x = \frac{z+\bar{z}}{2}$ and $y=\frac{z-\bar{z}}{2\mathrm{i}}$ leading to the usual definitions of $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \bar{z}}$ and then those of $dz$ and $\mathrm{d}\bar{z}.$ But somehow this is quite unsatisfying. One possible way out is suggested in the book by John P. D'Angelo where he says that (p. 148) smooth real functions $g_1, \ldots, g_m$ form a coordinate system on an open subset $\Omega \subset \mathbb{R}^n$ if the function $g = (g_1, \ldots, g_m)$ is injective on $\Omega$ and $\mathrm{d}g_1 \wedge \mathrm{d}g_2 \wedge \ldots \wedge \mathrm{d}g_m$ is nonzero. That far everything is in complete harmony with the intuition based on the implicit function theorem. But from now on, things seem to get a bit too clumsy when he says:

1) "This concept makes sense when these functions are either real or complex valued. For example, the functions $z$ and $\bar{z}$ define a coordinate system on $\mathbb{R}^2,$ because $\mathrm{d}x + \mathrm{i} \ \mathrm{d}y$ and $\mathrm{d}x - \mathrm{i} \ \mathrm{d}y$ are independent and the map $(x,y) \mapsto (x + \mathrm{i} y, x - \mathrm{i}y),$ embedding $\mathbb{R}^2$ into $\mathbb{C}^2,$ is injective." I am not sure if I got it right: does this really mean that, quite generally, the $g_i$'s are complex functions of real arguments with the differentials linearly independent over $\mathbb{R}$ (and not over $\mathbb{C}$)? I mean: I can see that this is precisely the case but: why should that be? To put it differently: where the linear independence over $\mathbb{C}$ comes into play and does it make sense at all? Related questions are:

2) Why is the last part of the formula (34) expressing $\mathrm{d}f,$ $$ \sum_{j=1}^{n} \frac{\partial f}{\partial x_j}\mathrm{d}x_j + \sum_{j=1}^{n} \frac{\partial f}{\partial y_j} \mathrm{d}y \stackrel{?}{=} \sum_{j=1}^{n} \frac{\partial f}{\partial z_j}\mathrm{d}z_j + \sum_{j=1}^{n} \frac{\partial f}{\partial \bar{z}_j} \mathrm{d}\bar{z}, $$ true and how do I get it?

3) And finally, are these considerations to suggest that, somehow, the complex analytic functions are precisely those $f(z, \bar{z})$ that do not depend on $\bar{z}$? If so, in exactly what sense is that statement claimed? Is there any baby complex manifolds theory involved?



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