understanding of $d(\log f(z))$ in complex analysis 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? 

 A: 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.
A: [Partial answer] 


*

*Consider the case when $f(z)=z$ and $\gamma$ is the unit circle $\{z:|z|=1\}$.  Note that the function $\frac{1}{z}$ does not have an antiderivative
on $\mathbb{C}\backslash\{0\}$, or even on a neighbourhood of the
unit circle (or any other curve going around the origin). Therefore,
one cannot interpret the formula $ \frac{1}{z}\ dz=d\log(z) $ as $$
   \frac{d\log(z)}{dz}=\frac{1}{z}, \quad z\in\gamma.  $$

*Therefore, one should really think (1.1) as written in a "formal" way. (Conway pointed out in his Functions of One Complex Variable that one cannot define $\log f(x)$ on $\gamma$ since otherwise one would have $\int_\gamma f'/f=0$, which is nonsense.) On the other hand, $\int_\gamma d\ln|z|$ is defined in Chapter III of Gameline's book as  aline integral:
$$
\int_\gamma d\ln|z|=\int_\gamma d\big(\frac{1}{2}\log(x^2+y^2)\big)
=\int_\gamma \frac{x}{x^2+y^2}dx+\frac{y}{x^2+y^2}dy
$$

*Another way to look at the integral $\int_\gamma\frac{f'(z)}{f(z)}\ dz$ without using the annoying discussion of differentials is via change of variables:
$$
\frac{1}{2\pi i}\int_\gamma\frac{f'(z)}{f(z)}\ dz=\frac{1}{2\pi i}\int_{f\circ\gamma}\frac{1}{z}\ dz\tag{*}
$$
the right hand side of which is defined to be the winding number of the (closed) curve $f\circ\gamma$ around $z=0$. If ${f\circ\gamma}$ is homotopic in ${{\mathbb C} \backslash \{0\}}$ (as a closed curve, up to reparameterisation) to a curve of the form ${t \mapsto z_0 + r e^{imt}}$, ${t \in [0,2\pi]}$, then $(*)$ can be calculated by Cauchy's theorem. 
