In Gameline's Complex Analysis Chapter 8, the notation $d(\log f(z))$ is used:
Here are my questions:
In the real case, suppose for any $x\in\mathbb{R}$, one has $f(x)\neq 0$ and $f$ is differentiable. Then one has $$ d(\log(f(x))=\frac{f'(x)}{f(x)}dx $$ by the chain rule.
But in the complex case, if the $\log$ function is not differentiable on the curve $\gamma$, how should one makes sense of $d\log f(z)$?
Similarly, how should one understand $d\arg(z)$?
$fdx+gdy$ is exact if $dh=fdx+gdy$ for some $h$ according to Gameline's book. But what does "$d\arg(z)$ is not exact" mean?
If $f:\>\Omega\to{\mathbb R}$ or $f:\>\Omega\to{\mathbb C}$ is a "bona fide" function defined on some domain $\Omega\subset{\mathbb C}$ its differential $df$ is a "closed one-form", and can be expanded as $$df=\quad adx+bdy,\quad g dz+h d\bar z,\quad gdz\ ,$$ depending on the context. Closed one-form means that the integral $\int_\gamma df$ has the same value $f(q)-f(p)$ for all curves beginning at $p$ and ending at $q$, and is $=0$ for all closed curves.
Now already in Calculus 102 we write $d\phi$ when working with polar coordinates, even though the variable $\phi$ is not a bona fide function in the punctured plane $\dot{\mathbb R}^2$ or $\dot{\mathbb C}$. Usually one glosses over this point and "integrates from $\phi=0$ to $\phi=2\pi$". It is however true that each point $z_0\in \dot{\mathbb C}$ has a pretty large neighborhood in which one can select a well defined branch of the variable $\phi$ that serves all desired purposes. This implies that the one-form denoted by $d\phi$ is exact. By the way, it expands as $$d\phi={-y\over x^2+y^2}dx+{x\over x^2+y^2}dy\ ,\tag{1}$$ whereby the right hand side is uniquely defined in all of $\dot{\mathbb R}^2$.
It is in this "abuse de language" sense that you have to understand the $d$ used in typographical assemblies like $d\log$ or $d{\rm arg}$. Corresponding to $(1)$ one has $$d\log(z)={dz\over z}\ ,$$ whereby the right hand side is well defined on all of $\dot{\mathbb C}$, and $d{\rm arg}$ is nothing else but the $d\phi$ alluded to above.