Building Rational Functions From Poles And Roots

In this blog entry I attempt to explain how you can build a polynomial up from its roots. That is the points where the function equals zero. I then explain how this same technique can be used to build rational functions, and what this means in the complex plane. This is something I use to generate a lot of the abstract mathematical art that I share on Tumblr, Flickr, and Deviant Art. So if you are interested in duplicating my art style, or just reading the rest of my blog, this entry is essential.

For sake of brevity, I will assume you have a working understanding of complex numbers. These are numbers of the form (a + b i) where (i) is defined to be the square root of negative one. Complex numbers are an entire topic to themselves. I may decide to do a crash course on them at some point, but until then there are plenty of great resources available online.

Take a look at the first function. This is a simple degree-one polynomial, also known as a monomial, or a line. It is trivial to see where f(z) is equal to zero. When z equals r1 the terms cancel and the result is zero, thus r1 is a root of our polynomial. In the second equation we have the product of two monomials. Because the product of two polynomials is always another polynomial, this whole expression is a polynomial. Notice also that when the first monomial is zero the whole expression is zero, since anything times zero is zero. This means that both r1 and r2 are roots of our polynomial.

We can extend this idea to build polynomials with as many roots as we like. We can even include certain roots more than once, increasing the multiplicity of that root. You may remember from algebra class, having to factor polynomials to determine their roots. This is sort of the same idea, but in reverse. We know what the roots are, and we want the polynomial.

More importantly, thanks to the fundamental theorem of algebra, we know that any polynomial of degree n has exactly n complex roots, although some roots may be doubles. This means that the polynomials we construct will have only the roots we specify, and no others. That way we can 'paint' a complex function simply by specifying where the function should be zero.

Image-1

​Here we have a complex polynomial with three roots. This image uses a simple domain coloring, where the output of the function determines the color at each point. In this case, the hue is derived from the argument of the function, while the brightness corresponds to the absolute value. The roots are the points that are completely black.

Because our polynomial contains only these roots and no others, they determine how the rest of our image looks. Notice how a ring of color surrounds every one of our roots. If we, for example, decided to double up on one of our roots, then the ring of color around that root would go through every hue twice. As it turns out, there is only one way to have the hue transition smoothly from one root to the next, and that determines the image we generate. Pretty neat, huh?

The astute of you might have noticed that our derived polynomial is not unique. In fact, infinitely many polynomials can be said to share the same roots. After all, we can always multiply our function by a constant, and it will still have the same roots. Once again, anything times zero is still zero. However, multiplying by a constant only changes the magnitude of the output, and not the argument. This bodes well for us, as we can adjust this extra parameter to control the 'brightness' of our image.

But why stop here? If we can build polynomials, we can also build rational functions. A rational function is any function that can be written as the ratio of two polynomials. Its a bit like a rational number, but for functions. All we have to do is specify the roots for the top and bottom polynomials, also know as the numerator and denominator, just like we normally would. But what does this mean for our combined function?

The numerator is easy enough to understand. When the numerator is zero, our function equates to zero divided by some value. This is almost always zero. This means that the roots of our top polynomial are usually roots of our combined function as well. We will get to the exception to this rule in just a little bit.

The denominator is a bit tricky. As you have probably had hammered into you from a young age, you cannot divide by zero. In the real world, we say that the function is undefined when q(z) equals zero. The roots of the denominator generally lead to asymptotes when you plot the real valued function on a graph. However, we aren’t concerned with the real world, and when you enter the complex plane interesting things start to happen.

Image-2

Hear is another function with the same domain coloring as before. Once again, the black spots mark where the function is equal to zero. But what about those white spots? These are the spots where our denominator equals zero. Just like the roots, they also have rings of color around them. However these colors process in the opposite direction from the roots. They are sort of like anti-roots and are referred to as 'poles' within complex analysis.

Poles and zeros share a duality. You can see that the color changes smoothly across the poles with no discontinuity. In a certain sense, one can say the value of the function at a pole is infinity, not some positive or negative infinity, but a singular point at infinity. If that sounds weird, consider wrapping the real number line back upon itself so that it forms a circle. If you do that with the complex plane you end up with a sphere, known as the Riemann sphere.

Image Licensed by Wikimedia Commons

Image Licensed by Wikimedia Commons

This is not as crazy as it sounds. There are several geometric reasons to support the Riemann sphere. Consider a line as a real function. The slope of the line is given as it's rise over run. As you rotate the line the slope increases. When the line turns vertical the slope is infinite. When you rotate it some more, you pass infinity and the slope becomes negative. In the case of our complex function, you can invert the function and the roots and poles swap places.

In this realm, division by zero always results in infinity, except for zero divided by zero, which is still indeterminate. So what happens when you have a root and a pole at the same point? Technically that point becomes a discontinuity. However, the function usually remains perfectly smooth around the point, and the point itself is infinitely small so we don't even see it. So effectively the root and the pole just cancel each other out. This is not entirely rigorous, and ideally you would check the limit.

One last note. Some may be tempted to think that because we used two polynomials in the definition of our rational function, we would have two extra terms to consider. As we have shown before, our definition of a polynomial is only unique up to a constant value. However, this results in multiplying both the top and bottom by a constant. The ratio of those two constants is itself a constant, so effectively we can treat this as a single constant.

Image-3

So that's how you can build your own polynomials and rational functions, just by specifying the roots and the poles. By placing the roots and poles in strategic locations, and fiddling with the domain coloring, you can create all sorts of interesting designs. I originally got into abstract mathematics because I didn’t like math teachers telling me I couldn’t do something. To me math is an art form, and is as much about creativity as it is rigorous proof. That said, you have got to know how the rules work, so that you can know how to bend them.