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Topic: Z^n and the Riemann Sphere (Read 7505 times) |
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Icarus
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Z^n and the Riemann Sphere
« on: Dec 19th, 2002, 9:07pm » |
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Since no-one seems to be contributing to this forum. I thought I would point out a few things. A lot of the "trippy" behavior in the graphs can be traced to two things:
1) The behavior of Zn, and
2) Noise.
1) If you write a complex number in polar form: Z=ReiT, it is fairly easy to see the effects of raising it to the power n: Zn = RneinT. Look at the affects on the magnitude and the phase separately.
If you raise a positive real number to a power greater than 1, it moves away from 1: numbers greater than 1 become even greater, and numbers less than 1 become even smaller. If the power is less than 1, then the opposite effect is seen: numbers move closer to 1. The best way to see the corresponding effect on complex numbers is use the Riemann sphere.
To form the Riemann sphere, consider the complex plane as a plane in space, and add a sphere of radius 1 centered at 0, so the circle z=1 forms its equator. Now do a stereographic projection of the plane onto the sphere from the north pole: For each complex number z, the line through the north pole and z intersects the sphere in one other point. That point is the projection of z onto the sphere, and is the Riemann sphere representation of z. This leaves one point left over on the sphere: the north pole. Since the closer you get to the pole, the farther out is the corresponding number on the plane, it make sense to define the north pole to be "infinity" (oo). 0 on the sphere is now the south pole, while the circle z=1 remains the equator.
A lot of complex analysis makes more sense when it is done on the Riemann sphere instead of the complex plane. For instance, a complex function f(z) has a pole at z0 if it is analytic around z0 and, when viewed on the Riemann sphere, limz->z0f(z) = oo. (An essential singularity is one in which the limit does not exist in any sense.) Thus, on the Riemann sphere, we can consider a pole as simply another value of the function.
Several processes have natural interpretations on the Riemann sphere. Negation is the reflection of the point through the sphere's axis (the diameter of the sphere running through 0 and oo). Conjugation is the reflection through the real plane (the plane on which the circle of the real numbers lies), and the multiplicative inverse of a point is its reflection through the equitorial plane. The antipode (reflection through the origin) is the opposite of the inverse.
Now examine the effect of raising z to the power n on the magnitude. When |n| > 1, everything moves away from the equator z=1 and towards the poles. The amount of movement is the same both above and below the equator. Where as on the plane large numbers are move great distances by raising them to a positive n, but numbers near 0 become slightly closer to zero, on the sphere the action is entirely symmetric. Similarly, if |n|<1, the points all move away from the poles and closer to the equator. When n < 0, the two hemispheres trade places, but otherwise the action is the same.
The effect on raising to a power on phase is easy to see: the phase angle just gets multiplied by n. Thus if z is allowed to move around a circle of constant magnitude, z2 will move around its circle twice, z3 three times, etc. On the other hand z1/2 runs into problems: once z makes it all the way back to the starting point, z1/2 is half way around from the value it started with. This forces z1/2 to have a discontinuity. Where this discontinuity occurs, however, is a personal choice: as long as the region you are looking at does not completely incircle 0, you can always put the discontinuity outside it. Similar remarks apply to other fractional or irrational powers.
Now compare the graphs of w=f(z) and w=f(zn). For n>1, activities of f(z) that take place far from the equator will be moved closer to the equator for f(zn). Seen from "above", the radial effect will look like someone took the geography of f(z) and scrunched it up around the equator, stretching it out at the poles. The greater the value of n, the more pronounced the effect will by. For 0 < n < 1, the reverse effect is seen: scrunching at the poles and stretching at the equator. If n < 0, the hemispheres will be reversed as well. There will also be a 180o rotation, but that combines with the phase effect. Since z2 goes around the plane/sphere twice for each trip of z around it, the geometry of f(z) gets laid down twice by f(z2).
Suppose you are the angel in charge of laying down the geography on new worlds. God has settled on a design He really likes and decided that all future worlds will receive it. In the world factory, the geography comes in a huge repeating sheet. You attach the edge of sheet to a meridian of the world and let the world turn pulling the sheet onto itself. When the starting meridian comes around again, you cut off the geography just at the point it starts to repeat, then set up the next world. Just when you've got the process down the plant clown Luc doubles the rate at which the geography scrolls out without you noticing. Suddenly you find yourself looking a world with two Mt Everests, two Amazon Jungles, two Grand Canyons, all 180o apart from their copies. This is the phase effect of f(z2).
But things can get worse. you quietly place this world in the out pile, hoping that quality control will not look too close, and reset your machine. Just then Luc comes by again, and as you bawl him out for the cruel trick you fail to notice that he now halves the rate of geography flow. You set up the next world and set it turning. To your horror, you find that the Dead sea regions are butting up a line in the Mid-Pacific. You have no choice but to cut off the geography there, and watch the ensuing cataclysm as the ocean pours into the Dead Sea basin. Your job at the world factory is not going to survive this one. This is the effect of f(z1/2).
w=[f(z)]n exaggerates or dampens the effects of f(z) but does not move them around. For n > 1, peaks are taller, valleys are deeper (measured from "sea level" of 1). The phase effects suffer a similar exaggeration.
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| « Last Edit: Feb 23rd, 2003, 11:16am by Icarus » |
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