Analytic continuation of the logarithm

Question

This is an example from Serge Lang Complex Analysis book. In XI, 1. Continuation along a curve, it says

Let us start with the function $\log(z)$ defined by the usual power series on the disc $D_0$ which is centered at $1$ and has radius $<1$ but $>0$. Let the path be the circle of radius $1$ oriented counterclockwise as usual. If we continue $\log(z)$ along this path, and let $(g,D)$ be its continuation, then near the point $1$ it is easy to show that \begin{equation} g(z)=\log(z)+2\pi i. \end{equation} Thus $g$ differs from $f_0$ by a constant, and is not equal to $f_0$ near $z_0=1$.

I can't get the expression of $g$ that Lang says "It's easy to show". I tried constructing explicitly a sequence $(f_0,D_0),...,(g,D_0)$ which is the analytic continuation of $(f_0,D_0)$ but I had no success.

Answer 1

Essentially what is happening is that you have ended up at a different point on the associated Riemann Surface, which can be visualized by a spiral.

In more concrete terms, consider $$ \log(e^{i\alpha})\quad \text{ for } \alpha \in [0,2\pi)$$ Which is the values we get for the logarithm as we travel around the unit circle. If we decide to follow the natural convention and have $\log(e^{i\cdot0}) = \log(1) =0,$ then we can start considering the rotated values, and note that $\log(e^{i(2\pi-\epsilon)}) = i(2\pi-\epsilon) \to 2\pi i$ as we let $\epsilon \to 0.$

Answer 2

I believe the following answer is more appropriate, the previous answer talks about Riemann Surfaces and is too sketchy to my taste.

Let $D_0 = D_4$ be the right half plane, $D_1$ be the upper half plane, $D_2$ be the left half plane and $D_3$ be the lower half plane. These five open subsets of $\mathbb{C}$ have the following properties:

1. They are all simply connected,
2. For all $k = 0, 1, 2, 3$, $D_k \cup D_{k + 1}$ is also simply connected.

Consider the path $\gamma: [0, 2\pi] \to \mathbb{C}$ given by $\gamma(t) = \exp(2\pi i t)$ which is the circle of radius $1$ oriented counterclockwise, as usual. For $k = 0, 1, 2, 3, 4$, set \begin{align*} f_k: D_k &\to \mathbb{C} \\ f_k(z) &= \int_{\gamma(k/4)}^z\frac{1}{w}dw + \int_{\gamma|_{[0, k/4]}}\frac{1}{w}dw. \end{align*} By property 1., these functions are well defined and by property 2., for all $k = 0, 1, 2, 3$, $f_{k + 1}$ agrees with $f_k$ in $D_k \cap D_{k + 1}$. Therefore, $f_4 = f_0 + 2\pi i$ is the analytic continuation of $f_0 = \log$ along $\gamma$.