Review: Branches, Branch Cuts, and Riemann Sheets

The square root function and its branches are defined as follows:

(i)

The square root function $\lambda ^{1/2}$

Let $\lambda=\vert\lambda \vert e^{i\theta}$ . A function is defined by giving its formula and specifying its domain. The square root function $\lambda ^{1/2}$ is defined by

$$\lambda^{1/2} =\vert\lambda \vert^{1/2} e^{i\theta/2}~~~~~~~~~~\textrm{(Formula)}$$

where

$$\theta ~\textrm{is ~any~real~number~~~~~~~~~~~~~(Domain)}$$

(ii)

The first branch of $\lambda ^{1/2}$ .

This function, denoted by $\sqrt{\lambda}$ , is defined by

$$\sqrt{\lambda} =\vert\lambda \vert^{1/2} e^{i\theta/2}~~~~~~~~~~\textrm{(Formula)}$$

where

$$0\le \theta <2\pi ~\textrm{~~~~~~~~~~~~~(Domain)}$$

More succinctly, we have

$$1^{st}~ \textrm{branch~of~} \lambda^{1/2} \equiv \sqrt{\lambda}$$

$$~ = \vert \lambda \vert^{1/2}e^{i\theta/2}\quad 0\le \theta <2\pi$$

See Figure 4.10

Figure 4.10: The first branch of the map $\lambda ^{1/2}$ maps the complex $\lambda$ -plane onto the upper half of the complex $k$ -plane

(iii)

The second branch of $\lambda ^{1/2}$ .

This function, denoted by $-\sqrt{\lambda}$ , is obtained by restricting the domain of $\lambda ^{1/2}$ to $2\pi \le \theta <4\pi$ . In other words, $-\sqrt{\lambda}$ is defined by

$$-\sqrt{\lambda} =\vert\lambda \vert^{1/2} e^{i\theta/2}~~~~~~~~~~\textrm{(Formula)}$$

where

$$2\pi\le \theta <4\pi ~\textrm{~~~~~~~~~~~~~(Domain)}$$

Equivalently, the second branch is defined by

$$-\sqrt{\lambda} =-\vert\lambda \vert^{1/2} e^{i\theta'/2}~~~~~~~~~~\textrm{(Formula)}$$

where

$$0\le \theta' <2\pi ~\textrm{~~~~~~~~~~~~~(Domain)}$$

See Figure 4.11

Figure 4.11: The second branch of the map $\lambda ^{1/2}$ maps the complex $\lambda '$ -plane onto the lower half of the complex $k$ -plane

(iv)

The Riemann sheets of $\lambda ^{1/2}$ .

The two branches $\sqrt{\lambda}$ and $-\sqrt{\lambda}$ are the two components of the single function $\lambda ^{1/2}$ whose domain consists of two copies of the complex $\lambda$ -plane. The points along the real $\lambda$ -axis are glued together (i.e. identified) in the manner depicted in Figure 4.12.

We are forced to accept these two copies if one writes the image of $\lambda$ as

$$[\lambda]^{1/2}=\alpha +i \beta$$

in the $k$ -plane of Figure 4.12. Then the first Riemann sheet consists of the set of $\lambda$ 's for which $\beta>0$ , while the second Riemann sheet consists of those $\lambda$ 's for which $\beta<0$ . These sheets are joined continuously along their positive $x$ -axes by the requirement that the function

$$[\lambda]^{1/2} =\vert \lambda \vert^{1/2} e^{i\theta/2}%~~~~~~~~0\le\theta<4\pi$$ (459)

be continuous for all values of $\theta$ . As a result of this requirement the two sheets are joined in the manner depicted in Figure 4.12.

It is quite true that, by itself, the complex number $\lambda$ does not tell whether $\lambda$ is on the first or the second Riemann sheet. This information is found neither in the real nor in the imaginary part of $\lambda$ . Instead, it is inferred from the square root function, Eq.(4.59) Indeed, one has

$$0<\theta<2\pi ~~\Longrightarrow~~\lambda \in \textrm{1st~Riemann~sheet}$$

$$2\pi<\theta<4\pi ~~\Longrightarrow~~\lambda \in \textrm{2nd~Riemann~sheet}$$ (460)

The set of points $\theta =2\pi$ lies on both Riemann sheets; so does the set of points $\theta=0$ . However, these two sets are distinct: they lie on opposite sides of the branch cut of the upper or the lower $\lambda$ -plane.

Figure 4.12: The map $\lambda ^{1/2}$ whose domain consists of the upper and the lower Riemann sheets and whose range is the the complex $k$ -plane. The circle in the first Riemann sheet gets mapped into a semi-circle in the upper $k$ -plane, and the cirle in the second Riemann sheet gets mapped into a semi-circle in the lower $k$ -plane. A pair of points one above the other in the two Riemann sheets gets mapped into a pair of diametrically opposite points in the $k$ -plane. The points common to the two Riemann sheets get mapped into the real axis in the k-plane.

An alternative way of characterizing the square root function is to collapse its domain, the two Riemann sheets, into a single $\lambda$ -plane. Such a simplification comes, however, at a price: the square root function is now two-valued, it has two formulas, the two branches $\sqrt{\lambda}$ and $-\sqrt{\lambda}$ . The domain for both of them is the $\lambda$ -plane with a branch cut along the positive real axis across which each branch is discontinuous.