Is there any single variable complex analysis textbook which uses $\textrm{d}\bar{z}$?
Every single variable text I have found defines what a complex line integral with respect to $\textrm{d}z$ means, and also $\textrm{d}s$, but none of them defines an integral with respect to $\textrm{d}\bar{z}$, or $\textrm{d}\bar{z} \wedge \textrm{d} z$.
I want to be able to write computations like
$$ \begin{align*} \int_{d\Omega} \frac{f(z)}{z-\zeta} \textrm{d}z &= \iint_\Omega d\left( \frac{f(z)}{z-\zeta}\textrm{d}z \right)\\ &= \iint_\Omega \frac{\partial f}{\partial \bar{z}} \frac{1}{z-\zeta} \textrm{d}\bar{z} \wedge\textrm{d}z \\ \end{align*} $$
without needing to convert everything back into $\textrm{d}x$ and $\textrm{d}y$ terms. No textbook I have uses these symbols at all, much less develops how to do this kind of calculus.
Something I find to be completely crazy is that I have never seen the following computation in print anywhere that I can remember, even though it is surely one of the most fundamental computations in all of complex analysis:
$$ \begin{align*} \int_{S^1} \frac{1}{z} \textrm{d}z &= \int_{S^1} \bar{z} \textrm{d}z \textrm{, since $\frac{1}{z} = \bar{z}$ on $S^1$}\\ &= \iint_D \textrm{d}\bar{z} \wedge \textrm{d}z\\ &= \iint_D 2i\textrm{d}x \wedge \textrm{d}y \textrm{, since $\textrm{d}\bar{z} \wedge \textrm{d}z = 2i\textrm{d}x \wedge \textrm{d}y$}\\ &=2\pi i \textrm{, since the area of the unit circle is $\pi$} \end{align*} $$
I think this is also because the textbooks essentially ignore non-holomorphic functions and their exterior derivatives entirely.
I never felt like I really understood single variable complex analysis until I learned how to do these kinds of computations while learning several complex variables.
I am looking for such a book because I cannot really imagine teaching the subject without these tools.