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Last time we left off trying to think of shapes to draw on our input plane that would help us better understand our function f(z)=z^2+1.
Since the z squared means to multiply z by itself and z is a complex number,
our functions behavior should have some connection to complex multiplication.
Back in part 7 we saw that one way to interpret complex multiplication is a rotation and scaling of our input values.
When we multiply two complex numbers together, their magnitudes multiplied and their angles added.
So the z squared part of our function should take our complex number z square its distance to the origin and double its angle.
The plus one portion of our equation is a little less exciting,
adding a positive real number will move all of our points in the positive real direction,
so to the right in this case by one.
Since this shift to the right doesn't affect the behavior we're interested and we'll leave it out of the equation for now.
Let's test the idea that our function will double the angle of its input values.
What kind of shape should we draw to test this idea?
Ideally we want to draw the shape that is made up of points that are all at the same angle,
to see if our function changes all points of the same angle in the same way.
So what kind of shape is made up of points all of the same angle?
This turns out to be a straight line through the origin.
If we draw a line like this in our input space,
we see that the output is also a straight line and what looks like double the angle.
We can add a few more lines to confirm this trend.
So we've shown that our transformation doubles the angle of our input values.
Now what about the magnitude of our inputs?
We said earlier that when squaring a complex number, the magnitude of the complex number should also be squared.
So the distance to the origin from our input points should be squared in our mapping.
What kind of shape should we draw to test this idea?
We want to test magnitude alone, so it would be nice to have a shape with a constant magnitude.
What kind of shape has the same magnitude or distance to the origin everywhere?
The shape we need is exactly a circle.
If we add sections of a few circles to our picture, we see that these results in new circle sections, but now at different distances to the origin.
So as we expected, our circles are preserved by their radii are changed.
Now we're really getting somewhere, by carefully choosing our input shape, we were able to better understand exactly what our function does two complex numbers.
Wonderful!
But before we celebrate here, let's keep a couple things in mind.
For one, we're dealing with a really simple function, and secondly, even for this simple function we can still run into trouble with our two complex plane set up.
For example, we know that our mapping doubles the angle of the input points.
This is fine until we use up too much of our input space.
If we continue the circles we started earlier, once we arrive at a hundred and eighty degrees, we begin to see a problem.
Our shapes have been expanded to fill the entire output space, but we've only used half of the input space.
As we continue our circles are new points have nowhere to go, except directly on top of our old points.
This does make sense algebraically because of the way squaring works. Points like (1+i) and (-1-i) will map the same exact output value namely 2i.
So it's not that our function is broken or anything, it's just that the technique we're using to visualize it can't really handle multiple values being mapped to the same location on the output plane.
After all, which pixels should we display, the one from here or the one from there?
Mappings like this create problems in mathematics, although typically in the reverse direction.
The reverse of a function where the inputs become outputs and the outputs become inputs is called the inverse.
The inverse of our function is pretty straightforward to find.
We simply need to solve our equation for z.
We now run into the real math problem.
Our inverse function represents the same exact connections are forward function, just in the opposite direction.
So our two inputs that map to the same output are now a single input that maps to two outputs.
The w value of 2i maps to both z=(1+i) and z=(-1-i).
This mapping from a single input to multiple outputs is a big enough problem that our functions inverse is not technically even considered a function.
The definition of a function requires that each input be mapped to one and only one output.
To make things nice and confusing and non functions like this are called multifunctions.
So our mapping is the same in either direction, but taken from z to w is considered a function, from wz is considered a multifunction, which isn't really a function.
Let's experiment with our multi-function:
when we draw shapes on rw-plane, our shapes are duplicated and shrunk down onto our z-plane.
Our shapes are copied because each two points in w is mapped to points and z and shrunk, because the square root function takes the square root of the magnitude of rw values, and divide each angle by two.
Let's experiment with one more type of shape, a path.
We'll pick a starting point on rw-plane wander around for a while and return where we started.
By following paths on both rw and z planes, we can get some idea what happens as we wander around the four-dimensional space of our complex multifunction.
Our first path returns right back to where we started on both z and w plane.
No surprise there.
But if we change our path a little, something weird happens.
RW path returns to where we started, but rz paths don't.
RZ values jump to a whole new part of the plane.
Somehow we've wondered our way into a completely new part of our multi-function.
So it seems that some paths on w lead us back to where we started, but others don't.
What could be going on here?
How is the complex landscape of our multifunction taking such similar paths in such different directions?
One reason i like math is that for many problems, someone much smarter than me, has already given them some serious thoughts, and quite often found an elegant solution.
In this case that person was one of Gauss's students Bernhard Riemann.
We'll discuss the solution next time.
If you're still watching, here's a quick bonus feature.
A very slick way of visualizing functions of complex variables is to do this.
This technique is called domain coloring, and it's only a couple of decades old.
By following where the various colors from our color wheel are mapped, we can learn a tremendous amount about how are complex function works.
For example we can really get a feel here for how the square root function unfolds rw-plane, into two copies of itself.
If we follow a path around the z-plane, we encounter every color twice.
There's a ton of cool and way more detailed things we can learn from these domain coloring plots, and the wikipedia article is a great place to start.
Ok, the video is actually on stop button.
Seriously stop...
... for real...
... this has gone on way too long and is awkward for both of us...
... stop.