# An analytic continuation of the square root along the unit circle

I want to find an analytic continuation of the square root along the unit circle but I am not sure whether I am doing it correctly.

Let $$C_0$$ be the open disk of radius $$1$$ around $$1$$, and let $$f_0:C_0 \to \mathbb{C}$$ be defined as $$f_0(re^{i \varphi})=\sqrt{r} e^{i \frac{\varphi}{2}}$$, where $$\varphi \in (-\pi,\pi]$$. Let $$\gamma: [0,1] \to \mathbb{C}$$ be the path given $$\gamma(t)=e^{2 it \pi}$$. Find an analytic continuation of $$f_0$$ along $$\gamma$$, i.e. a sequence $$(C_k,f_k)_{k=0}^{n}$$ of analytic continuations $$f_k$$ of $$f_0$$ such that the $$C_k$$ cover the image of $$\gamma$$. Show that $$f_n(1)=-f_0(1)$$.

I tried to do this as follows. Since $$f_0$$ is holomorphic on $$C_0$$ we have the power series expansion

$$f_0(z)=\sum_{m=0}^{\infty} a^{(0)}_m (z-1)^m \tag{1}$$

where $$a^{(0)}_m=\frac{1}{m!} \frac{\partial^m}{z^m} \sqrt{z} \big|_{z=1}$$. I wanted to analyticaly continue this series by defining $$C_1=\{z \in \mathbb{C} \ | \ |z-e^{i \frac{\pi}{4}}|<1 \}$$ and considering the function

$$f_1: C_1 \to \mathbb{C}, \ f_1(z)=\sum_{m=0}^{\infty} a^{(1)}_m (z-e^{i \frac{\pi}{4}})^m.$$

where $$a^{(1)}_m=\frac{1}{m!} \frac{\partial^m}{z^m} \sqrt{z} \big|_{z=e^{i \frac{\pi}{4}}}$$ and $$arg(z) \in (-\frac{3 \pi}{4}, \frac{5 \pi}{4}]$$. Let $$z=re^{i\varphi} \in C_0 \cap C_1$$. With $$z_1=e^{i \frac{\pi}{4}}$$ I have

$$\sqrt{z}=\sqrt{z_1} \sqrt{\frac{z}{z_1}} =\sqrt{z_1} \sqrt{1+\frac{z}{z_1}-1} \underset{(1)}{=}\sqrt{z_1} \sum_{m=0}^{\infty} a^{(0)}_m (\frac{z}{z_1}-1)^m =\sum_{m=0}^{\infty} \frac{\sqrt{z_1}}{z^m_1} a^{(0)}_m (z-z_1)^m$$

Since the series representation of $$f_1$$ is unique and since $$f_0(z)=\sqrt{z}$$ the functions $$f_0,f_1$$ agree on $$C_0 \cap C_1$$. By iterating the steps above I can define disks $$C_2, C_3,...C_8$$ with centers $$e^{i k\frac{\pi}{4}}$$ and corresponding holomorphic functions $$f_k$$, $$k=2,...,8$$, each time requiring $$arg(z) \in (-\pi+k \frac{\pi}{4},\pi+k \frac{\pi}{4}]$$ for $$z \in C_k$$. Upon considering the power series expansion centered at $$e^{i \frac{8 \pi}{2}}=e^{i 2 \pi}$$ I should get $$f_8(e^{i 2 \pi})=\sqrt{1} e^{i \pi}=-1=-\sqrt{1} e^{i \cdot 0}=-f_0(1)$$. Am I on the right track here or is there an error?

• $$f_0 (re^{i\theta}) = \sqrt{r} e^{i\theta/2}$$ on $$\Omega_0 = \{re^{i\theta} \in \mathbb{C} \mid r> 0, \theta \in (-\pi/2, 3\pi/4)\}$$,
• $$f_1(re^{i\theta}) = \sqrt{r}e^{i\theta/2}$$ on $$\Omega_1 = \{ re^{i\theta} \in \mathbb{C} \mid r>0, \theta \in (\pi/2, 7\pi/4) \}$$,
• $$f_2(re^{i\theta}) = \sqrt{r}e^{i\theta/2}$$ on $$\Omega_2 = \{ re^{i\theta} \in \mathbb{C} \mid r>0, \theta \in (3\pi/2, 5\pi/2)\}$$.
These $$f_i$$ are analytic on their domains of definition, $$f_i, f_{i+1}$$ agree on the intersections of their domains, and $$f_2(1) = e^{i\pi} = -1 = - f_0(1)$$.