Rigorous meaning of the expression $dz = dx + idy$ Many complex analysis books just define $dz$ by $dz = dx + idy$.
In smooth manifold theory, the expressions like $dx$, $dy$, $df$ have precise meaning: covector field.
My question is:
What is the precise meaning of the expression '$dz = dx + idy$'? Do we understand this expression in terms of linear functionals on tangent space?

Recall if we interpret $x$ and $y$ are coordinates for $\mathbb{R}^2$, then $dx$ and $dy$ signify a basis for $T^* \mathbb{R}^2$. Since we view $T^* \mathbb{R}^2$ as a real vector space, any expression of the form $\alpha dx + \beta dy$ where $\alpha, \beta \in \mathbb{R}^2$ makes sense. But when doing complex analysis, or studying complex manifolds, one actually needs expressions of the form $dx + i dy$ which are meaningless if your vector space is real. The way to overcome this is to complexify your vector space. So if $V$ is a real vector space, we write $V \otimes \mathbb{C}$ as the complexified version of $V$. A simple way to interpret it is just to say that taking complex linear combinations is now legal. More precisely, if your original space had a basis $\{ dx,dy\}$ the complexified space also includes $\{i dx, i dy\}$ in the basis set, so the dimension of your space gets multiplied by $2$ if you view it as a real vector space, or viewing it as a complex vector space $\{dx, dy\}$ is still the basis, since you now your take your underlying field to be $\mathbb{C}$. In complex manifold theory, one always complexifies the tangent and cotangent spaces.
