TITLE: Cauchy transform
QUESTION [6 upvotes]: I'm trying to solve the problem: 
If $\phi \colon S_1(0)\to \Bbb C$ is a continuous function on the unit circle, its
Cauchy transform
$$C\phi(z) =\frac 1{2\pi i}\int_{S_1(0)}\frac{\phi(w)}{w-z}dw$$
is analytic on $\Bbb C \setminus S_1(0)$.
If $\phi$ is holomorphic on closure of $B_1(0)$ then it follows from Cauchy's formula that
$$C\phi(z) = \begin{cases}\phi(z)& \mbox{if }z \in B_1(0);\\0&\mbox{if }z \in C\setminus B_1(0).\end{cases}$$
What happens if $\phi(z) =\bar z$?
Hint: On the unit circle $\bar z = 1/z$. I tried to substitute in the integral by $\phi(w)=1/w$ and solve the integral by writing it as $(-1/zw + 1/z(w-z))$. I got the integral is $\log$, but I'm not sure at all of my solution. Can anyone please help me? 
Also, I'm trying to read more in Cauchy transform but I couldn't find any resource with the same definition. The books that I found have other definition, can anyone please give me more information about this transformation to read with the same definition that I have here? 
Thank you in advance.

REPLY [2 votes]: So far so good, $\int\displaystyle\frac1wdw=\log w$, but what is $\log$? It is the 'inverse' of $\exp$, though $\exp$ is not at all a bijection, it is a $\mathbb Z$-fold cover to $\mathbb C\setminus\{0\}$, periodic with $2\pi i$ period, as $e^{2\pi i}=1$. So, the $\log$ is unique only up to $2\pi ki$ for some $k\in\mathbb Z$.
If you calculate the integral only on the upper semicircle, from angle $0$ to $\pi$, you will get $\pi i$. Similarly, for the other semicircle, so
$$\int_{S_1(o)}\frac1wdw = 2\pi i$$
However, if you calculate the integral of $\log$ on a circle that doesn't surround the pole (the denominator, now zero), then the result is $0$.