Multivalued Functions : I
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First of all, watch the following video, which covers the content presented here. Although this page does stand on its own, you’ll get much more out of it after seeing the video.
The square root function is known as a multivalued function, because you cannot unequivocally say which number was squared to give a certain number. That is, I could have squared either 2 or -2, and, either way, I'd have gotten 4. The question "what's the square root of 4" actually has two answers. This video shows the corresponding square root (on the left) as we walk along a path in the complex domain (on the right). You'll notice that when we return to every point the second time, it has a different square root than the first time we saw it. This animation starts with choosing 2 as the square root of 4, but as we circle around and return, the square root has only circled halfway: it is -2. Walking around again returns us to 2:
Riemann developed a new way of visualizing complex functions, one that allowed the double nature of the square root to be an inherent quality of the space it operates within. The “Riemann surfaces” (as they are now known) that he developed, transform what appears to be paradoxical behavior in the standard complex plane (the perceptual projection) into natural behavior in the true (dynamic) space of any function. By looking at what the square-root function does, we see that it implicitly inhabits the action space we see below: it has two layers. We can understand the function by the space it lives in. Riemann showed that different functions inhabit (or, create) different spaces, even though their effects all project onto the standard perceptual space of the complex plane, just as the wide variety of different physical principles determining actions around us are invisible to us, although their effects reach us through the senses.
Here is the Riemann surface for the square-root:
Click here for an interactive version, showing the square roots above, and the squares below. By using the sliders, you can move the path. Note that if the path doesn't include the center point (the "origin"), the two-layered nature does not appear.
A point around which a function takes different values is called a "branch-point", and the different sheets are known as branches. For our square-root example, zero (the origin) is called a branch-point of order two. Another way of viewing why it's called a branch point comes from taking this view of a path. As we walk along the complex plane, and pass through zero, the square root of our value has to split. That is, let's say we start at 4 and take its positive square root, 2 (as opposed to -2), and then walk to the origin, passing through it. What should the “answer” be once we have negative numbers? When we get to -1, there are two equally good answers, +i and -i. As the path comes to zero, which direction should it go in? Neither direction is a more correct continuation than the other. In fact, the path cannot be continued in an single way in this square-root mapping. The path splits in two: it "branches."
Riemann developed a new way of visualizing complex functions, one that allowed the double nature of the square root to be an inherent quality of the space it operates within. Look at the same animation, but now from the side:
For any root, the power of the root is the number of branches, which are related to each other by the roots of unity. That is, the two answers for the square root always differ from each other by -1, or 180°. There are two ways to see this. One is that any number and its negative have the same square: (-z)2 = (-1 · z)2 = (-1)2z2 = (+1)z2, because -1·-1 = 1. The other is to think of squaring the 180° action. Performing that action twice moves you 360°, putting you back in the same place. So, two complex numbers, 180° apart, give the same number when squaring. If you look at this path, you'll see that the two points corresponding to same squared number are exactly opposite each other.
And here, for the cube roots, the three positions on the left (separated by 120°) correspond to the same position on the right. As before, since 120° is a third of a full rotation, taking any number and rotating it 120°, and then cubing the original value as well as the rotated one, gives the same result, since the 120° rotation will be tripled. Unlike the previous visualization of square roots, where the two positions on the left made the same square on the right, we now have a Riemann surface on the right, and can imagine the three different locations as being distinct, although they'd look the same as seen from the top.
This means that the three sheets on the right differ from each other by that 120° rotation. A function (mapping) can include multiple such branch-points. For example, adding a cube root and a square root (with different origins) creates a total of six sheets, with three branchings around the cube root and two around the square root. Every point on the complex plane can stand in three different relations with respect to the cube root, and two with respect to the square root. (Animation not yet finished.)
Higher Branch Points
In addition to such branch points, Riemann also introduces branch points with an infinite number of sheets. To take a common example, think of the sine function. As we move around the circle, we can measure our progress by the distance along the circumference we have travelled. If we'd like to convert this distance into the usual x and y directions, we have the sine function for the vertical part and the cosine for the horizontal part. If we measure the angle as the distance along the circumference, rather than the convenient, but somewhat arbitrary 360°, then we'd say that a full rotation is 2π for a circle of radius 1. Rather than degrees, this angle measurement is known as radians, since we're measuring rotation as how many times the radius of the circle we have traversed the circumference.
These repeat themselves even as the point runs laps around the closed circular track. So if I said, “I'm thinking of an angle whose sine is ½,” there are many possible answers: 30°, 390°, 750°, and so on. The number of answers really goes on forever. Think also of a wall clock. If I wanted to leave a room, and have the hour hand move 1/12 of the dial when I came back, I could either return in 1 hour, 13 hours, 25 hours, 37 hours, etc. There are not a finite, but an infinite number of possible answers, unlike the square root, which only had two. The black square we walked on in the square-root function took us around twice, but then we were back at our original spot. With the sine or the hour hand on a clock, you just keep going forever.
I’d like to show you the Riemann surface for the sine function, but it is not as simple as you’d think. That's because Riemann surfaces are complex, and it's hard to think of the sine of the rotation +i, as opposed to 120°. Instead, we’ll use a related function: the complex exponential. (If the next paragraph doesn’t work for you, just imagine the sine when we get to the next animation.)
This special function, the exponential of the natural logarithm base (e), includes the sine and the cosine. We have ex+iy = ex(cos y + isin y), which means that it repeats itself every time y increases by a full rotation (2π), or when its argument (x+iy) increases by 2πi. So, to answer “e raised to what power equals a given number,” you'd get an infinite number of answers, all differing by 2πi. For example, the answer to the question ez=1, has as its answers for z: 0, 2πi, 4πi, etc. (If you’re not familiar with e, jump to 21:50 in this video to get some background.)
Here you can see the repeated, periodic motion of the complex logarithm. As we repeat the same circle on the right, the logarithm of the number continues to change without repeating.
The complex natural exponential (or, rather, its inverse, the complex natural logarithm), has not two sheets like the square root, but rather an infinite number of branches. Here you have the Riemann surface view of the complex natural logarithm:
Next, we'll look at other ways of visualizing the characteristics of these functions.