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8. Mathematical Functions
When introducing complex algebra in Chapter 4, 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.
8.1 Non-integer powers as multi-valued operations
Given a complex number in its polar representation, z = reiθ^ , raising to the power of p could be handled this way:
zp^ =
reiθ^
)p = rpeipθ^. (8.1)
Let’s take a closer look at the complex exponential term eipθ^. Since θ = arg(z) is an angle, we can change it by multiples of 2π without altering the value of z. So we can re-write the above equation as
zp^ =
r ei(θ+2πn)
)p
rpeipθ^
e^2 πinp, where n ∈ Z. (8.2)
If p is an integer, this re-statement is somewhat trivial: no matter what integer n we take, 2 πnp is a multiple of 2π, so zp^ ends up with the same value:
zp^ = rpeipθ^ unambiguously (if p ∈ Z). (8.3)
But if p is not an integer, the exp (2πinp) factor takes on different values for different n. In that case, 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.
8.2 Roots of unity
Let’s take a closer look at the problematic exponential term,
exp (2πinp) , where n ∈ Z. (8.4)
If p is irrational, 2πnp never repeats 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. Any rational number 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 πnp = 2πn
P
Q
= 2π ×
Q
Q
(Q − 1)
Q
(modulo 2π). (8.5)
The set of values is independent of the numerator P. The value of P merely affects the sequence in which the numbers are generated as we step though the integer values of n. This is demonstrated by the following examples:
Example—Consider the complex square root operation, z^1 /^2. If we write z in its polar respresentation, z = reiθ^ , then
z^1 /^2 =
[
r ei(θ+2πn)
] 1 / 2
= r^1 /^2 eiθ/^2 eiπn, n ∈ Z. (8.6)
The eiπn^ factor has two possible values: +1 (for even n) and −1 (for odd n). Hence, the values of the square root are
z^1 /^2 = r^1 /^2 eiθ/^2 × { 1 , − 1 }. (8.7)
Example—Consider the cube root operation z^1 /^3. Taking z = reiθ^ , we obtain
z^1 /^3 = r^1 /^3 eiθ/^3 e^2 πin/^3 , n ∈ Z. (8.8)
Running through n gives
n · · · − 2 − 1 0 1 2 3 4 · · · 2 πn/ 3 · · · − 4 π/ 3 − 2 π/ 3 0 2 π/ 3 4 π/ 3 6 π/ 3 8 π/ 3 · · · 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 · · ·
Hence, 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 taking z = reiθ^ ,
z^2 /^3 = r^2 /^3 e^2 iθ/^3 e^4 πin/^3 , n ∈ Z. (8.10)
Running through n gives
n · · · − 2 − 1 0 1 2 3 4 · · · 4 πn/ 3 · · · − 8 π/ 3 − 4 π/ 3 0 4 π/ 3 8 π/ 3 12 π/ 3 16 π/ 3 · · · 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, z^2 /^3 = r^2 /^3 e^2 iθ/^3 ×
1 , e^2 πi/^3 , e−^2 πi/^3
Note that the set of values in curly brackets is the same as in the previous example, demonstrating that the numerator P does not affect the set.
From the above examples, we deduce the following expression for rational powers:
zP/Q^ = rP/Q^ eiθ^ (P/Q)^ ×
1 , e^2 πi/Q, e^4 πi/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 z along the branch cut. In other words, we will only consider complex numbers whose polar representation can be written as
z = reiθ^ , θ ∈ (−π, π).
(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 root of unity +1. On this branch, for z = reiθ^ , the value is f+(z) = r^1 /^2 eiθ/^2 , θ ∈ (−π, π).
- 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).
The following plot shows how varying z affects the positions of f+(z) and f−(z) in the complex plane:
Branch cut
In the left subplot, the red dashes indicate the branch cut, and the various symbols (circle, square, star, and triangle) indicate representative values of z. In the right subplots, the symbols indicate the corresponding positions of f+(z) and f−(z) in the complex plane. 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 complex plane (i.e., the branch cut is excluded). It can moreover be shown that these functions are analytic over all of the complex plane except the branch cut (see Section 6.2); 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 8.4.3.
8.4.2 Different branch cuts for the complex square root
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). (8.19)
Next, we use the same formulas as before to define the branches of the complex square root:
f±(z) = ±r^1 /^2 eiθ/^2. (8.20)
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 values produced by the branch functions are shown in the following figure:
Branch cut
The 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. We can think of the branch cut as a boundary where two branches are “glued” together, so that crossing the branch cut brings us from one branch to a different branch. For example, in the left subplot, consider the value of z indicated by the triangle, which lies just above the branch cut. In the right subplots, observe that the corresponding value of f+(z) lies just above the positive real axis, and f−(z) lies just below the negative real axis. Next, consider the value of z indicated by the star, which lies just below the branch cut. Going from the triangle to the star is equivalent to a small downwards displacement of z, “crossing” the branch cut. Now the values of the positive and negative branches are swapped: f−(z) lies just below the positive real axis, near where f+(z) was previously, and f+(z) now lies just above the negative real axis where f−(z) was previously. The three-dimensional plot below provides another way to visualize the role of the branch cut. Here, the horizontal axes correspond to x = Re(z) and y = 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. The choice of branch cut, shown as a red line, is just a choice about how to divide up the branches of a multi-valued operation.
8.6 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 (i) every branch point has a branch cut ending on it, and (ii) every branch cut ends on a branch point. 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.
8.6.1 An important example
We can illustrate the process of assigning branch cuts, and defining branch functions, with the following 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 the branch points at z = ±1. 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 , (8.23) z − 1 = r 2 eiθ^2 , (8.24)
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). (8.25)
Hence, f (z) can be written as
f (z) = ln
r 1 r 2
- i(θ 1 − θ 2 + 2πm), where
m ∈ Z, 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 us 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, depending on Re[z]: Firstly, for Re[z] < −1 (to the left of the leftmost branch point),
Im[z] = 0+^ ⇒ θ 1 → π, θ 2 → π ⇒ f (z) = ln
r 1 r 2
(π) − (π)
= ln
r 1 r 2
Im[z] = 0−^ ⇒ θ 1 → −π, θ 2 → −π ⇒ f (z) = ln
r 1 r 2
(−π) − (−π)
= ln
r 1 r 2
(If you’re not sure why θ 1 and θ 2 have these values, look carefully at the above figure, and think about what values θ 1 and θ 2 would have for, say, z = −2 + 0. 001 i or z = − 2 − 0. 001 i.) 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+^ ⇒ θ 1 → 0 , θ 2 → π ⇒ f (z) = ln
r 1 r 2
(0) − (π)
= ln
r 1 r 2
− iπ
(8.29)
Im[z] = 0−^ ⇒ θ 1 → 0 , θ 2 → −π ⇒ f (z) = ln
r 1 r 2
(0) − (−π)
= ln
r 1 r 2
(8.30)
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+^ ⇒ θ 1 → 0 , θ 2 → 0 ⇒ f (z) = ln
r 1 r 2
= ln
r 1 r 2
Im[z] = 0−^ ⇒ θ 1 → 0 , θ 2 → 0 ⇒ f (z) = ln
r 1 r 2
= ln
r 1 r 2
8.7 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
