By the time of 2019 and my sixth A-to-Z series , I had some standard narrative tricks I could deploy. My insistence that everything is polynomials, for example. Anecdotes from my slight academic career. A prose style that emphasizes what we do with the idea of something rather than instructions. That last comes from the idea that if you wanted to know how to compute a Taylor series you’d just look it up on Mathworld or Wikipedia or whatnot. The thing a pop mathematics blog can do is give some reason that you’d want to know how to compute a Taylor series. I regret talking about functions that break Taylor series, though. I have to treat these essays as introducing the idea of a Taylor series to someone who doesn’t know anything about them. And it’s bad form to teach how stuff doesn’t work too close to teaching how it does work. Readers tend to blur what works and what doesn’t together. Still, is a really neat weird function and it’d be a shame to let it go completely unmentioned.
Today’s A To Z term was nominated by APMA, author of the Everybody Makes DATA blog. It was a topic that delighted me to realize I could explain. Then it started to torment me as I realized there is a lot to explain here, and I had to pick something. So here’s where things ended up.
In the mid-2000s I was teaching at a department being closed down. In its last semester I had to teach Computational Quantum Mechanics. The person who’d normally taught it had transferred to another department. But a few last majors wanted the old department’s version of the course, and this pressed me into the role. Teaching a course you don’t really know is a rush. It’s a semester of learning, and trying to think deeply enough that you can convey something to students. This while all the regular demands of the semester eat your time and working energy. And this in the leap of faith that the syllabus you made up, before you truly knew the subject, will be nearly enough right. And that you have not committed to teaching something you do not understand.
So around mid-course I realized I needed to explain finding the wave function for a hydrogen atom with two electrons. The wave function is this probability distribution. You use it to find things like the probability a particle is in a certain area, or has a certain momentum. Things like that. A proton with one electron is as much as I’d ever done, as a physics major. We treat the proton as the center of the universe, immobile, and the electron hovers around that somewhere. Two electrons, though? A thing repelling your electron, and repelled by your electron, and neither of those having fixed positions? What the mathematics of that must look like terrified me. When I couldn’t procrastinate it farther I accepted my doom and read exactly what it was I should do.
It turned out I had known what I needed for nearly twenty years already. Got it in high school.
Of course I’m discussing Taylor Series. The equations were loaded down with symbols, yes. But at its core, the important stuff, was this old and trusted friend.
The premise behind a Taylor Series is even older than that. It’s universal. If you want to do something complicated, try doing the simplest thing that looks at all like it. And then make that a little bit more like you want. And then a bit more. Keep making these little improvements until you’ve got it as right as you truly need. Put that vaguely, the idea describes Taylor series just as well as it describes making a video game or painting a state portrait. We can make it more specific, though.
A series, in this context, means the sum of a sequence of things. This can be finitely many things. It can be infinitely many things. If the sum makes sense, we say the series converges. If the sum doesn’t, we say the series diverges. When we first learn about series, the sequences are all numbers. , for example, which diverges. (It adds to a number bigger than any finite number.) Or , which converges. (It adds to .)
In a Taylor Series, the terms are all polynomials. They’re simple polynomials. Let me call the independent variable ‘x’. Sometimes it’s ‘z’, for the reasons you would expect. (‘x’ usually implies we’re looking at real-valued functions. ‘z’ usually implies we’re looking at complex-valued functions. ‘t’ implies it’s a real-valued function with an independent variable that represents time.) Each of these terms is simple. Each term is the distance between x and a reference point, raised to a whole power, and multiplied by some coefficient. The reference point is the same for every term. What makes this potent is that we use, potentially, many terms. Infinitely many terms, if need be.
Call the reference point ‘a’. Or if you prefer, x0. z0 if you want to work with z’s. You see the pattern. This ‘a’ is the “point of expansion”. The coefficients of each term depend on the original function at the point of expansion. The coefficient for the term that has is the first derivative of f, evaluated at a. The coefficient for the term that has is the second derivative of f, evaluated at a (times a number that’s the same for the squared-term for every Taylor Series). The coefficient for the term that has is the third derivative of f, evaluated at a (times a different number that’s the same for the cubed-term for every Taylor Series).
You’ll never guess what the coefficient for the term with is. Nor will you ever care. The only reason you would wish to is to answer an exam question. The instructor will, in that case, have a function that’s either the sine or the cosine of x. The point of expansion will be 0, , , or .
Otherwise you will trust that this is one of the terms of , ‘n’ representing some counting number too great to be interesting. All the interesting work will be done with the Taylor series either truncated to a couple terms, or continued on to infinitely many.
What a Taylor series offers is the chance to approximate a function we’re genuinely interested in with a polynomial. This is worth doing, usually, because polynomials are easier to work with. They have nice analytic properties. We can automate taking their derivatives and integrals. We can set a computer to calculate their value at some point, if we need that. We might have no idea how to start calculating the logarithm of 1.3. We certainly have an idea how to start calculating . (Yes, it’s 0.3. I’m using a Taylor series with a = 1 as the point of expansion.)
The first couple terms tell us interesting things. Especially if we’re looking at a function that represents something physical. The first two terms tell us where an equilibrium might be. The next term tells us whether an equilibrium is stable or not. If it is stable, it tells us how perturbations, points near the equilibrium, behave.
The first couple terms will describe a line, or a quadratic, or a cubic, some simple function like that. Usually adding more terms will make this Taylor series approximation a better fit to the original. There might be a larger region where the polynomial and the original function are close enough. Or the difference between the polynomial and the original function will be closer together on the same old region.
We would really like that region to eventually grow to the whole domain of the original function. We can’t count on that, though. Roughly, the interval of convergence will stretch from ‘a’ to wherever the first weird thing happens. Weird things are, like, discontinuities. Vertical asymptotes. Anything you don’t like dealing with in the original function, the Taylor series will refuse to deal with. Outside that interval, the Taylor series diverges and we just can’t use it for anything meaningful. Which is almost supernaturally weird of them. The Taylor series uses information about the original function, but it’s all derivatives at a single point. Somehow the derivatives of, say, the logarithm of x around x = 1 give a hint that the logarithm of 0 is undefinable. And so they won’t help us calculate the logarithm of 3.
Things can be weirder. There are functions that just break Taylor series altogether. Some are obvious. A function needs lots of derivatives at a point to have a good Taylor series approximation. So, many fractal curves won’t have a Taylor series approximation. These curves are all corners, points where they aren’t continuous or where derivatives don’t exist. Some are obviously designed to break Taylor series approximations. We can make a function that follows different rules if x is rational than if x is irrational. There’s no approximating that, and you’d blame the person who made such a function, not the Taylor series. It can be subtle. The function defined by the rule , with the note that if x is zero then f(x) is 0, seems to satisfy everything we’d look for. It’s a function that’s mostly near 1, that drops down to being near zero around where x = 0. But its Taylor series expansion around a = 0 is a horizontal line always at 0. The interval of convergence can be a single point, challenging our idea of what an interval is.
That’s all right. If we can trust that we’re avoiding weird parts, Taylor series give us an outstanding new tool. Grant that the Taylor series describes a function with the same rule as our original function. The Taylor series is often easier to work with, especially if we’re working on differential equations. We can automate, or at least find formulas for, taking the derivative of a polynomial. Or adding together derivatives of polynomials. Often we can attack a differential equation too hard to solve otherwise by supposing the answer is a polynomial. This is essentially what that quantum mechanics problem used, and why the tool was so familiar when I was in a strange land.
Roughly. What I was actually doing was treating the function I wanted as a power series. This is, like the Taylor series, the sum of a sequence of terms, all of which are times some coefficient. What makes it not a Taylor series is that the coefficients weren’t the derivatives of any function I knew to start. But the experience of Taylor series trained me to look at functions as things which could be approximated by polynomials.
This gives us the hint to look at other series that approximate interesting functions. We get a host of these, with names like Laurent series and Fourier series and Chebyshev series and such. Laurent series look like Taylor series but we allow powers to be negative integers as well as positive ones. Fourier series do away with polynomials. They instead use trigonometric functions, sines and cosines. Chebyshev series build on polynomials, but not on pure powers. They’ll use orthogonal polynomials. These behave like perpendicular directions do. That orthogonality makes many numerical techniques behave better.
The Taylor series is a great introduction to these tools. Its first several terms have good physical interpretations. Its calculation requires tools we learn early on in calculus. The habits of thought it teaches guides us even in unfamiliar territory.
And I feel very relieved to be done with this. I often have a few false starts to an essay, but those are mostly before I commit words to text editor. This one had about four branches that now sit in my scrap file. I’m glad to have a deadline forcing me to just publish already.
Thank you, though. This and the essays for the Fall 2019 A to Z should be at this link. Next week: the letters U and V. And all past A to Z essays ought to be at this link.