# Branch points of $f(z)= \frac{\sqrt{z} \log(z)}{(1+z)^2}$

How does one go about finding the branch points/holomorphic branches of a multi-function composed of several other multi-functions? Here is an example of what I mean:

Let $$f(z)= [\frac{\sqrt{z} \log(z)}{(1+z)^2}]$$ be a multifunction. Identify the branch points and find a holomorphic branch.

I have no idea how to approach this if $$\sqrt{z}$$ and $$\log(z)$$ are considered multi-functions themselves. Can someone help me out here? Thanks!

• I like the question. I guess: since $\sqrt{z} = e^{(\log z)/2}$, the whole thing has only the branch behavior of $\log z$. – GEdgar Jan 10 '20 at 13:37
• @GEdgar So everything comes down to a choice of branch of $\log(z)$? – user489116 Jan 10 '20 at 13:38
• Another way to say it: $\sqrt{z}$ is single-valued on the Riemann surface of $\log z$. But I await an actual answer from someone who knows more about this! – GEdgar Jan 10 '20 at 13:41

We have two multi-valued functions $$\sqrt{z}$$ and $$\log{z}$$ to consider when looking for branch points of \begin{align*} \frac{\sqrt{z}\log{z}}{(1+z)^2}\tag{1} \end{align*}

We start by looking at each of the functions separately to recall the base information. At first the simpler one:

Branch points of $$\sqrt{z}$$:

Since $$\sqrt{z}$$ is the inverse of the function $$z^2$$ we expect $$w(z)=\sqrt{z}$$ having two values. To analyse the multivaluedness it is convenient to use polar coordinates and we write $$z=re^{i\theta}$$ with $$\theta=\theta_{p}+2\pi n$$ and $$0\leq \theta_{p}<2\pi, n\in\mathbb{Z}$$. We then have \begin{align*} w=r^{1/2} e^{i\theta_{p}/2}e^{n\pi i}\tag{2} \end{align*} where $$r^{1/2}=\sqrt{r}\geq 0$$ and $$n$$ is an integer. For a given value of $$z$$, the function $$w(z)=\sqrt{z}$$ takes two possible values corresponding to $$n$$ even and $$n$$ odd, namely \begin{align*} \sqrt{r}e^{i\theta_{p}/2}\qquad\text{and}\qquad \sqrt{r}e^{i\theta_{p}/2}e^{\pi i}=-\sqrt{r}e^{i\theta_{p}/2} \end{align*}

• The key observation here is, when traversing a small circuit around $$z=0$$, we do not return to the original value. When starting for instance at $$z=\varepsilon$$ for real $$\varepsilon >0$$ and letting $$n=0$$ we start with $$\theta_p=0$$ and after returning to $$z=\varepsilon, \theta_p=2\pi$$ we have according to (2) $$w=\sqrt{\varepsilon}e^{2i\pi/2}=-\sqrt{\varepsilon}$$. Traversing another circuit we have $$w=\sqrt{\varepsilon}e^{4i\pi/2}=\sqrt{\varepsilon}$$ and this continues for $$n$$ even and $$n$$ odd.

Since a point is a branch point of a function $$w(z)$$ iff it is discontinuous upon traversing a small circuit around this point, we have $$0$$ identified as branch point of the function $$w(z)=\sqrt{z}$$.

• The point at $$z=\infty$$ is another branch point of $$\sqrt{z}$$ which can be seen as follows. We consider $$z=\frac{1}{t}$$ and note that $$0$$ is a branch point of $$t^{-1/2}$$ which can be argued in the same we as we did above for $$z^{1/2}$$.

• If we take a closed circuit that does not enclose $$z=0$$ then the phase $$\theta_p$$ varies continuously between two values $$0\leq \theta_1\leq \theta_p\leq \theta_2<2\pi$$ as $$z$$ traverses the circuit returning exactly to its previous value with no phase change.

We are now in the situation to restrict $$w$$ to an open region of the plane so that the resulting function is single-valued and continuous. One way to do so is by taking $$n=0$$ and restricting the region by cutting out the real positive axis and by also deleting $$z=0$$ and $$z=\infty$$. The semiaxis $$\Re z>0$$ is called the branch cut.

Branch points of $$\log{z}$$:

Since $$\log{z}$$ is the inverse of the function $$e^z$$ which has period $$2\pi i$$ we expect $$w(z)=\log{z}$$ having countably infinite values. We use polar coordinates and write $$z=re^{i\theta}$$ with $$\theta=\theta_{p}+2\pi n$$ and $$0\leq \theta_{p}<2\pi, n\in\mathbb{Z}$$. We then have \begin{align*} w=\log{r} + i\theta_{p}+2n\pi i\tag{3} \end{align*} where $$\log r> 0$$ and $$n$$ is an integer. For a given value of $$z$$, the function $$w(z)=\log{z}$$ takes infinite values corresponding to $$n\in\mathbb{Z}$$.

• Again, when traversing a small circuit around $$z=0$$, we do not return to the original value. When starting for instance at $$z=\varepsilon$$ for real $$\varepsilon >0$$ and letting $$n=0$$ we start with $$\theta_p=0$$ and after returning to $$z=\varepsilon, \theta_p=2\pi$$ we have according to (3) $$w=\log{r} + i\theta_{p}+2\pi i$$. Traversing another circuit we have $$w=\log{r} + i\theta_{p}+4\pi i$$ and this continues for $$n\in \mathbb{Z}$$.

So, we have $$0$$ identified as branch point of the function $$w(z)=\log{z}$$.

• The point at $$z=\infty$$ is another branch point of $$\log{z}$$ which can be seen as follows. We consider $$z=\frac{1}{t}$$ and note that $$0$$ is a branch point of $$\log{\frac{1}{t}}=-\log{t}$$ which can be argued in the same way as we did above for $$\log{z}$$.

• If we take a closed circuit that does not enclose $$z=0$$ then the phase $$\theta_p$$ varies continuously between two values $$0\leq \theta_1\leq \theta_p\leq \theta_2<2\pi$$ as $$z$$ traverses the circuit returning exactly to its previous value with no phase change.

We are now in a similar situation as with $$w=\sqrt{z}$$. We have two branch points and can use the same branch cut $$\{z|z>0\}$$ to restrict $$w(z)=\log{z}$$ to an open region where $$w$$ is single-valued and continuous. We might also have cut out the real negative axis with phase angle $$\theta_p$$ in $$(-\pi,\pi]$$ and obtain this way the principal value of the logarithm.

Branch points of $$\frac{\sqrt{z}\log{z}}{(1+z)^2}$$:

We can proceed similarly as we did above. At first we note that it is sufficient to consider $$w(z)=\sqrt{z}\log{z}$$ since $$\frac{1}{(1+z)^2}$$ is analytic in $$\mathbb{C}\setminus\{-1\}$$ and single-valued. In order to analyse the multivaluedness of $$w(z)$$ we write $$z=re^{i\theta}$$ with $$\theta=\theta_{p}+2\pi n$$ and $$0\leq \theta_{p}<2\pi, n\in\mathbb{Z}$$. We then have \begin{align*} w=r^{1/2} e^{i\theta_{p}/2}e^{n\pi i}(\log{r} + i\theta_{p}+2n\pi i)\tag{4} \end{align*} where $$r\geq 0$$ and $$n$$ is an integer. We will see for a given value of $$z$$, the function $$w(z)=\sqrt{z}\log{z}$$ takes infinite values corresponding to $$n\in\mathbb{Z}$$.

• Indeed, when traversing a small circuit around $$z=0$$, we do not return to the original value. When starting for instance at $$z=\varepsilon$$ for real $$\varepsilon >0$$ and letting $$n=0$$ we start with $$\theta_p=0$$ and after returning to $$z=\varepsilon, \theta_p=2\pi$$ we have according to (4) $$w=\sqrt{\varepsilon}e^{2i\pi/2}\left(\log{r} + 2\pi i\right)=-\sqrt{\varepsilon}\left(\log{r} + 2\pi i\right)$$. Traversing another circuit we have $$w=\sqrt{\varepsilon}\left(\log{r} + 4\pi i\right)$$ and this continues for $$n\in \mathbb{Z}$$ changing the sign for $$n$$ even and $$n$$ odd.

Again, we have $$0$$ identified as branch point of the function $$w(z)=\sqrt{z}\log{z}$$.

• The point at $$z=\infty$$ is another branch point of $$\sqrt{z}\log{z}$$ which can be seen as above by substituting $$z=\frac{1}{t}$$ and considering $$t$$ close by $$0$$ which can be argued in the same way as we did above.

• If we take a closed circuit that does not enclose $$z=0$$ then the phase $$\theta_p$$ varies continuously between two values $$0\leq \theta_1\leq \theta_p\leq \theta_2<2\pi$$ as $$z$$ traverses the circuit returning exactly to its previous value with no phase change.

We are now in a similar situation as with the functions $$\sqrt{z}$$ and $$\log{z}$$. We have two branch points $$z=0$$ and $$z=\infty$$ and can use the same branch cut $$\{z|z>0\}$$ to restrict $$w(z)=\sqrt{z}\log{z}$$ to an open region where $$w$$ is single-valued and continuous.

Note: This answer follows closely section: 2.2 Multivalued functions from Complex Variables: Introduction and Applications by M.J. Ablowitz and A.S. Fokas.

• Very helpful! Thank you very much! – user489116 Jan 18 '20 at 11:36
• @user: You're welcome. Thanks a lot for accepting the answer and granting the bounty. :-) – Markus Scheuer Jan 18 '20 at 11:44