Branch cuts of $\log z$ Consider the function $\displaystyle f(z)=\log z$ which is multi-valued function, as $\displaystyle \log z =\log|z|+i\arg(z)+2n\pi i$ , where $n$ is an integer. Also I know that $f(z)$ is NOT analytic non-positive real axis. 
But in my book I found that : $f(z)$ is analytic and single valued when $|z|>0$ and $\alpha<\arg(z)<\alpha+2\pi$ for a given real number $\alpha$.

I am unable to understand that why $f$ is single valued and analytic in $\alpha<\arg(z)<\alpha+2\pi$ ? Please help me...

 A: The formula
$$\log z = \log |z| + i \arg(z) + 2 n \pi i, \quad\text{where $n$ is an integer}
$$
is a poor expression of the multivalued nature of the function $\log z$, and I suspect this is where your confusion arises. The point is that $\arg(z)$ is already multi-valued all on its own; throwing in "$+2 n \pi$" is entirely unnecessary and confuses the point. Here is a correct, multi-valued expression of $\arg(z)$ (expressed as a set):


*

*$\arg(z) = \{\theta \in \mathbb{R} \,\bigm|\, \cos(\theta) + i \sin(\theta) = z \, / \, |z|\}$


One could then observe that 


*

*For any $\theta \in \arg(z)$ one has $\arg(z) = \{\theta + 2 \pi n \,\bigm|\, n \in \mathbb{Z}$}


One can then write $\log(z)$ as a multivalued function on the domain $\mathbb{C}-0$:


*

*$\log(z) = \log|z| + i \arg(z)$


Finally, one can correctly say


*

*For any $a \in \mathbb{R}$, on the subset of $\mathbb{C} - \{0\}$ where $z \, / \, |z| \ne \cos(a) + i \sin(a)$ there is an analytic and single valued branch of $\log(z)$ where $\arg(z)$ is chosen in the interval $a < \arg(z) < a + 2 \pi$. (Notice: the "branch cut" in this situation is just the ray defined by the equation $z  \, / \, |z| = \cos(a) + i \sin(a)$.)


The reason this works is because if $z \, / \, |z| \ne \cos(a) + i \sin(a)$ then, for any choice of $\theta \in \arg(z)$, using the expression $\arg(z) = \{\theta + 2 n \pi \,\bigm|\, n \in \mathbb{Z}\}$, it follows that there exists a unique $n \in \mathbb{Z}$ such that $\theta + 2 n \pi \in (a,a+2\pi)$.
A: The function $g(z):={1\over z}$ is analytic in $\dot{\mathbb C}:={\mathbb C}\setminus\{0\}$. If $\Omega$ is  any simply connected domain contained in $\dot{\mathbb C}$ the function $g$  has an analytic primitive $$G:\quad \Omega\to{\mathbb C},\qquad z\mapsto G(z)$$ which can be defined, e.g., by choosing a $z_0\in\Omega$ and putting
$$G(z):=\int_{z_0}^z{1\over \zeta}\>d\zeta$$
and integrating along a curve $\gamma\subset\Omega$.
If you choose $\Omega$ by slitting ${\mathbb C}$ along the neagtive real axis and choose $z_0:=1$ you obtain the principal value of $\log$, denoted often by ${\rm Log}$. But you are free to slit ${\mathbb C}$ along any other ray ${\rm arg}(z)={\rm const.}$, or even along some more complicated curve to your liking.
