Branch of $\sqrt{1-z^2}$

Question

Show that a branch of $\sqrt{1-z^2}$ can be defined in any region $\Omega$ where the points $1,-1$ are in the same component of its complement.

This is a question in Ahlfors' Complex Analysis (P.148 Q5) that I came across while trying to self-study the book. I tried to tackle the problem by considering $\Omega=\mathbb{C} \backslash [-1,1]$ first, and tried the approach as in Section 4.4 Corollary 2, namely find a branch of the corresponding log first. For this $\Omega$, the image of $1-z^2$ is $\mathbb{C} \backslash [0,1]$, so a branch of $\log(1-z^2)$ cannot be defined; evidently one needs to construct the branch of $\sqrt{1-z^2}$ directly. Here is where I ran out of ideas... Any help is appreciated!

Answer 1

Let $\gamma$ be a closed curve in $\Omega$. Then $$ \int_\gamma \bigl( \frac{1}{z-1} - \frac{1}{z+1} \bigr) \, dz = 2 \pi i \bigl( n(\gamma, 1) - n(\gamma , -1) \bigr) $$ where $n$ denotes the winding number. $1$ and $-1$ are in the same component of $\mathbb C \setminus \Omega$ and therefore also in the same component of $\mathbb C \setminus |\gamma|$. Since the winding number is constant on each component of $\mathbb C \setminus |\gamma|$, the integral is zero.

This holds for all closed curves in $\Omega$. It follows that the integrand has a holomorphic antiderivative $$ F : \Omega \to \mathbb C, \quad F'(z) = \frac{1}{z-1} - \frac{1}{z+1} \, . $$ Now define $$ f : \Omega \to \mathbb C, \quad f(z) = (z+1)e^{F(z)/2} \, , $$ this is (almost) the holomorphic branch of $\sqrt{1-z^2}$.

The function $$ g : \Omega \to \mathbb C, \quad g(z) = \frac{f(z)^2}{z^2 -1} = \frac{z+1}{z -1} e^{F(z)} $$ satisfies $$ \frac{g'(z)}{g(z)} = \frac{1}{z+1} - \frac{1}{z-1} + F'(z) = 0 $$ It follows that $g$ is constant and therefore $$ f(z)^2 = C (1-z^2) $$ for some constant $C$. Then $f(z)/\sqrt{C}$ is a holomorphic branch of $\sqrt{1-z^2}$ in $\Omega$.

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The idea behind those calculations was to show that $ F(z) = \log \frac{z-1}{z+1} $ has a holomorphic branch in $\Omega$, then $$ e^{F(z)} = \frac{z-1}{z+1} = \frac{z^2-1}{(z+1)^2} $$ which leads to the "candidate" $f(z) = (z+1)e^{F(z)/2}$.

Answer 2

A branch of $\sqrt{1-z^2}$ is a lift of $f(z)=1-z^2$ with respect to the covering map $p:\mathbb{C}^\times\to\mathbb{C}^\times,\,p(z)=z^2 $. Fix a basepoint $z_0\in\Omega$, and let $\tilde{w_0}\in \mathbb{C}^\times$ be a point such that $p(\tilde{w_0})=f(z_0)$. By the theory of covering spaces, $f$ has a lift with respect to $p$ if and only if $f_\ast(\pi_1(\Omega,z_0))\subset p_\ast(\pi _1(\mathbb{C}^\times,\tilde{w_0}))$. Now we may identify $\pi_1(\mathbb{C}^\times,{f(z_0)})$ with $\mathbb{Z}$ via the integration $$\gamma\mapsto \frac{1}{2\pi i}\int_\gamma \frac{1}{z}dz,$$ and under this identification $\operatorname{im}p_\ast$ corresponds to $2\mathbb{Z}$. THerefore, we have $\operatorname{im}f_\ast\subset\operatorname{im}p_\ast$ if and only if $$\frac{1}{2\pi i} \int_{f\ast\gamma}\frac{1}{z}dz\in 2\mathbb{Z}$$ for (every piecewise $C^1$) curve. On the other hand, $$\frac{1}{2\pi i} \int_{f\ast\gamma}\frac{1}{z}dz=\frac{1}{2\pi i}\int_{\gamma}\frac{2z}{z^2-1}dz=\frac{1}{2\pi i}\int_\gamma\left(\frac{1}{1-z}+\frac{1}{1+z}\right)dz$$ is the sum of the winding number of $\gamma$ around $1$ and $-1$. By our hypothesis on $\Omega$, these winding numbers are equal, and we are done.

Answer 3

Here's an approach that directly uses the function $\frac{2z}{1-z^2}$, and without covering space theory.
Let $\gamma$ be a closed curve in $\Omega$. By partial fraction decomposition: $$\int_\gamma\frac{2z}{1-z^2}dz=-\int_\gamma\left(\frac{1}{z-1}+\frac{1}{z+1}\right)dz=-2\pi i(n(\gamma,1)+n(\gamma,-1)).$$ But $-1,1$ lie in the same component of $\Omega^c$, hence $n(\gamma,1)=n(\gamma,-1)$. Thus $$\int_\gamma \frac{2z}{1-z^2}dz=4\pi ik;\;k\in \mathbb{Z}.$$ Fix $a\in \Omega$, and define $\displaystyle F_\alpha(w)=\int_\alpha \frac{2z}{1-z^2}dz$, where $\alpha$ is a path from $a$ to $w$. By the above, any two paths $\alpha,\beta$ from $a$ to $w$ satisfy that $F_\alpha(w)=F_\beta(w)+4\pi ik$. Hence for any two paths $\alpha,\beta$, we have that $F'_\alpha=F'_\beta$ and $e^{F_\alpha}=e^{F_\beta}$. Fix a path $\alpha$ and write $F=F_\alpha$. It follows that the map $g(z)=e^{F(z)}(1-z^2)$ is well-defined. Moreover, so is $g'$, which is given by \begin{align*} g'(z) &= F'(z)e^{F(z)}(1-z^2)-2ze^{F(z)}\\ &= \frac{2z}{1-z^2}e^{F(z)}(1-z^2)-2ze^{F(z)}\\ &=0, \end{align*} whence $g(z)\equiv c$ for some $c\in \mathbb{C}$. Note that in particular $c\neq 0$, so we may rewrite $g(z)=e^{2c}$ for some $c\in \mathbb{C}$. Then $1-z^2=e^{-F(z)+2c}$. Lastly, define a branch of $\sqrt{1-z^2}$ via $\sqrt{1-z^2}=e^{-F_\alpha(z)/2+c}$ for some path $\alpha$ from $a$ to $z$. Recall that any other choice of path $\beta$ gives another value $F_\beta$, which differs from $F_\alpha$ by a multiple of $4\pi i$, so the value of $\sqrt{1-z^2}$ is independent of the choice of $\alpha$.