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7 Branch Points and Branch Cuts
When introducing complex algebra, we postponed discussion of what it means to raise a complex number to a non-integer power, such as z^1 /^2 , z^4 /^3 , or zπ^. It is now time to open that can of worms. This involves learning about the two indispensible concepts of branch points and branch cuts.
7.1 Non-integer powers as multi-valued operations
Given a complex number in its polar representation, z = r exp[iθ], raising to the power of p could be handled this way: zp^ =
reiθ^
)p = rpeipθ^. (1)
Let’s take a closer look at the complex exponential term eipθ^. Since θ = arg(z) is an angle, we can change it by any integer multiple of 2π without altering the value of z. Taking this fact into account, we can re-write the above equation more carefully as
zp^ =
r ei(θ+2πn)
)p
rpeipθ^
e^2 πinp^ where n ∈ Z. (2)
Thus, there is an ambiguous factor of exp(2πinp), where n is any integer. If p is an integer, there is no problem, since 2πnp will be an integer multiple of 2π, so zp^ has the same value regardless of n: zp^ = rpeipθ^ unambiguously (if p ∈ Z). (3)
But if p is not an integer, there is no unique answer, since exp (2πinp) has different values for different n. In that case, “raising to the power of p” is a multi-valued operation. It cannot be treated as a function in the usual sense, since functions must have unambiguous outputs (see Chapter 0).
7.1.1 Roots of unity
Let’s take a closer look at the problematic exponential term,
exp (2πinp) , n ∈ Z. (4)
If p is irrational, 2πnp never repeats itself modulo 2π. Thus, zp^ has an infinite set of values, one for each integer n. More interesting is the case of a non-integer rational power, which can be written as p = P/Q where P and Q are integers with no common divisor. It can be proven using modular arithmetic (though we will not go into the details) that 2πn (P/Q) has exactly Q unique values modulo 2π:
2 πn
P
Q
= 2π ×
Q
Q
(Q − 1)
Q
(modulo 2π). (5)
This set of values is independent of the numerator P , which merely affects the sequence in which the numbers are generated. We can clarify this using a few simple examples:
Example—Consider the complex square root operation, z^1 /^2. If we write z in its polar respresentation, z = reiθ^ , (6) then z^1 /^2 =
[
r ei(θ+2πn)
] 1 / 2
= r^1 /^2 eiθ/^2 eiπn, n ∈ Z. (7)
The factor of eiπn^ has two possible values: +1 for even n, and −1 for odd n. Hence,
z^1 /^2 = r^1 /^2 eiθ/^2 × { 1 , − 1 }. (8)
Example—Consider the cube root operation z^1 /^3. Again, we write z in its polar repre- sentation, and obtain z^1 /^3 = r^1 /^3 eiθ/^3 e^2 πin/^3 , n ∈ Z. (9) The factor of exp(2πin/3) has the following values for different n:
n · · · − 2 − 1 0 1 2 3 4 · · · e^2 πin/^3 · · · e^2 πi/^3 e−^2 πi/^3 1 e^2 πi/^3 e−^2 πi/^3 1 e^2 πi/^3 · · ·
From the pattern, we see that there are three possible values of the exponential factor:
e^2 πin/^3 =
1 , e^2 πi/^3 , e−^2 πi/^3
Therefore, the cube root operation has three distinct values:
z^1 /^3 = r^1 /^3 eiθ/^3 ×
1 , e^2 πi/^3 , e−^2 πi/^3
Example—Consider the operation z^2 /^3. Again, writing z in its polar representation,
z^2 /^3 = r^2 /^3 e^2 iθ/^3 e^4 πin/^3 , n ∈ Z. (12)
The factor of exp(4πin/3) has the following values for different n:
n · · · − 2 − 1 0 1 2 3 4 · · · e^4 πin/^3 · · · e−^2 πi/^3 e^2 πi/^3 1 e−^2 πi/^3 e^2 πi/^3 1 e−^2 πi/^3 · · ·
Hence, there are three possible values of this exponential factor,
e^2 πin(2/3)^ =
1 , e^2 πi/^3 , e−^2 πi/^3
Note that this is the exact same set we obtained for e^2 πin/^3 in the previous example, in agreement with the earlier assertion that the numerator P has no effect on the set of values. Thus, z^2 /^3 = r^2 /^3 e^2 iθ/^3 ×
1 , e^2 πi/^3 , e−^2 πi/^3
From the above examples, we deduce the following expression for rational powers:
zP/Q^ = rP/Q^ eiθ^ (P/Q)^ ×
1 , e^2 πi·(1/Q), e^2 πi·(2/Q),... , e^2 πi·[(1−Q)/Q]
The quantities in the curly brackets are called the roots of unity. In the complex plane, they sit at Q evenly-spaced points on the unit circle, with 1 as one of the values:
- Define a branch cut along the negative real axis, so that the domain excludes all values of z along the branch cut. In in other words, we will only consider complex numbers whose polar representation can be written as
z = reiθ^ , θ ∈ (−π, π). (21)
(For those unfamiliar with this notation, θ ∈ (−π, π) refers to the interval −π < θ < π. The parentheses indicate that the boundary values of −π and π are excluded. By contrast, we would write θ ∈ [−π, π] to refer to the interval −π ≤ θ ≤ π, with the square brackets indicating that the boundary values are included.)
- One branch is associated with the +1 root. On this branch, for z = reiθ^ , the value is
f+(z) = r^1 /^2 eiθ/^2 , θ ∈ (−π, π). (22)
- The other branch is associated with the root of unity −1. On this branch, the value is
f−(z) = −r^1 /^2 eiθ/^2 , θ ∈ (−π, π). (23)
In the following plot, you can observe how varying z affects the positions of f+(z) and f−(z) in the complex plane:
Branch cut
The red dashed line in the left plot indicates the branch cut. Our definitions of f+(z) and f−(z) implicitly depend on the choice to place the branch cut on the negative real axis, which led to the representation of the argument of z as θ ∈ (−π, π). In the above figure, note that f+(z) always lies in the right half of the complex plane, whereas f−(z) lies in the left half of the complex plane. Both f+ and f− are well-defined functions with unambiguous outputs, albeit with domains that do not cover the entire com- plex plane. Moreover, they are analytic over their entire domain (i.e., all of the complex plane except the branch cut); this can be proven using the Cauchy-Riemann equations, and is left as an exercise. The end-point of the branch cut is called a branch point. For z = 0, both branches give the same result: f+(0) = f−(0) = 0. We will have more to say about branch points in Section 7.2.3.
7.2.2 Different branch cuts for the complex square root
In the above example, you may be wondering why the branch cut has to lie along the negative real axis. In fact, this choice is not unique. For instance, we could place the branch cut along the positive real axis. This corresponds to specifying the input z using a different interval for θ: z = reiθ^ , θ ∈ (0, 2 π). (24)
Next, we use the same formulas as before to define the branches of the complex square root:
f±(z) = ±r^1 /^2 eiθ/^2. (25)
But because the domain of θ has been changed to (0, 2 π), the set of inputs z now excludes the positive real axis. With this new choice of branch cut, the branches are shown in the following figure.
Branch cut
These two branch functions are different from what we had before. Now, f+(z) is always in the upper half of the complex plane, and f−(z) in the lower half of the complex plane. However, both branches still have the same value at the branch point: f+(0) = f−(0) = 0. The branch cut serves as a boundary where two branches are “glued” together. You can think of “crossing” a branch cut as having the effect of moving continuously from one branch to another. In the above figure, consider the case where z is just above the branch cut. Then f+(z) lies just above the positive real axis, and f−(z) lies just below the negative real axis. Next, consider z lying just below the branch cut. This is equivalent to a small downwards displacement of z, “crossing” the branch cut. For this case, f−(z) now lies just below the positive real axis, near where f+(z) was previously. Moreover, f+(z) now lies just above the negative real axis, near where f−(z) was previously. Crossing the branch cut thus swaps the values of the positive and negative branches. The three-dimensional plot below provides another way to visualize the role of the branch cut. Here,the horizontal axes correspond to Re(z) and Im(z). The vertical axis shows the arguments for the two values of the complex square root, with arg
[
f+(z)
]
plotted in orange and arg
[
f−(z)
]
plotted in blue. If we vary the choice of the branch cut, that simply affects which values of the multi-valued operation are assigned to the + (orange) branch, and which values are assigned to the − (blue) branch. Hence, the choice of branch cut is just a choice about how to divide up the branches of a multi-valued operation.
this set, the magnitude goes to infinity as r → ∞. In this limit, the argument (i.e., the choice of root of unity) becomes irrelevant, and the result is simply ∞. By similar reasoning, one can prove that ln(z) has branch points at z = 0 and z = ∞. This is left as an exercise.
7.4 Branch cuts for general multi-valued operations
Having discussed the simplest multi-valued operations, zp^ and ln(z), here is how to assign branch cuts for more general multi-valued operations. This is a two-step process:
- Locate the branch points.
- Assign branch cuts in the complex plane, such that:
- Every branch point has a branch cut ending on it.
- Every branch cut ends on a branch point.
Note that any branch point lying at infinity must also obey these rules. The branch cuts should not intersect.
The choice of where to place branch cuts is not unique. Branch cuts are usually chosen to be straight lines, for simplicity, but this is not necessary. Different choices of branch cuts correspond to different ways of partitioning the values of the multi-valued operation into separate branches.
7.4.1 An important example
We can illustrate the process of assigning branch cuts, and defining branch functions, using the following nontrivial multi-valued operation:
f (z) = ln
z + 1 z − 1
This is multi-valued because of the presence of the complex logarithm. The branch points are z = 1 and z = −1, as these are the points where the input to the logarithm becomes ∞ or 0 respectively. Note that z = ∞ is not a branch point; at z = ∞, the input to the logarithm is −1, which is not a branch point for the logarithm. We can assign any branch cut that joins these two. A convenient choice is shown below:
This choice of branch cut is nice because we can express the z + 1 and z − 1 terms using the polar representations
z + 1 = r 1 eiθ^1 , (27) z − 1 = r 2 eiθ^2 , (28)
where r 1 , r 2 , θ 1 , and θ 2 are shown graphically in the above figure. The positioning of the branch cut corresponds to a particular choice for the ranges of the complex arguments θ 1 and θ 2. As we’ll shortly see, the present choice of branch cut corresponds to
θ 1 ∈ (−π, π), θ 2 ∈ (−π, π). (29)
Hence, in terms of this polar representation, f (z) can be written as
f (z) = ln
r 1 r 2
- i(θ 1 − θ 2 + 2πm), m ∈ Z,
where z = −1 + r 1 eiθ^1 = 1 + r 2 eiθ^2 , θ 1 , θ 2 ∈ (−π, π).
The choice of m specifies the branch, and we can choose m = 0 as the principal branch. Let’s now verify that setting θ 1 ∈ (−π, π) and θ 2 ∈ (−π, π) is consistent with our choice of branch cut. Consider the principal branch, and compare the outputs of the above formula for z just above the real axis, and for z just below the real axis. There are three cases of interest. Firstly, for Re(z) < 1 (to the left of the leftmost branch point),
Im(z) = 0+^ ⇒ f (z) = ln
r 1 r 2
(π) − (π)
= ln
r 1 r 2
Im(z) = 0−^ ⇒ f (z) = ln
r 1 r 2
(−π) − (−π)
= ln
r 1 r 2
Thus, there is no discontinuity along this segment of the real axis. Secondly, for − 1 < Re(z) < 1 (between the two branch points),
Im(z) = 0+^ ⇒ f (z) = ln
r 1 r 2
(0) − (π)
= ln
r 1 r 2
− iπ (33)
Im(z) = 0−^ ⇒ f (z) = ln
r 1 r 2
(0) − (−π)
= ln
r 1 r 2
Hence, in the segment between the two branch points, there is a discontinuity of ± 2 πi on different sides of the real axis. The value of this discontinuity is exactly equal, of course, to the separation between the different branches of the complex logarithm. Finally, for Re(z) > 1 (to the right of the rightmost branch point), there is again no discontinuity:
Im(z) = 0+^ ⇒ f (z) = ln
r 1 r 2
= ln
r 1 r 2
Im(z) = 0−^ ⇒ f (z) = ln
r 1 r 2
= ln
r 1 r 2
7.5 Exercises
- Find the values of (i)i. [solution available]
- Prove that ln(z) has branch points at z = 0 and z = ∞. [solution available]
- For each of the following multi-valued functions, find all the possible function values, at the specified z:
(a) z^1 /^3 at z = 1. (b) z^3 /^5 at z = i. (c) ln(z + i) at z = 1. (d) cos−^1 (z) at z = i
