In the second part of this series on series expansions (introduction), we found a way to make a series expansion using any basis of orthogonal polynomials. This is how the standard Fourier series is defined, and then we also used it to make a series expansion out of the Legendre polynomials. Now we’re going to compare how these behave with different numbers of terms.

This is finally the fun part of the seminar! My second-year physics students should generally have already known the previous two articles from first-year courses, so they started here, pretty much. To interact with these series expansions, you should load up my Jupyter notebook on either Colab or myBinder, and try plotting your own functions and seeing how the expansions behave.

In reality we don’t use the infinite form of series expansions, we only use the first few terms to get a good approximation of a function. The notebook is used for investigating how different series expansions work with comparably few terms, and which would be most appropriate in a given situation. The series there are the standard Fourier series, the Fourier–Legendre series, and the Taylor series about $x=0$ (which is formed by a different method to the other two). If you need a reminder of how these series look like, the Taylor and Fourier series forms were given in the introduction, and the Legendre series was derived in part two.

I asked the students to consider a couple of the following questions, and discuss them in groups of around five:

• Which series would you call the “most accurate”? Why? Does it depend on the function?
• What sorts of functions are the different expansions best at approximating? Which are they bad at?
• The Legendre series often seems to “give up” in the middle of some shapes at low orders (e.g. sinusoids). Why is this, and are there any things the series is still useful for?
• The Taylor series almost invariably has the largest pointwise error. Why is this? What is the Taylor series useful for?

In the first part of this series on series expansions (introduction), we defined several concepts from first-year linear algebra that we need to work in a general, abstract setting rather than having to repeat the same derivations over and over again. We are now looking at the topic at the centre of this series; generalising the Fourier series method.

Amazingly, the few definitions I gave in the previous article let us define a whole family of different functional series expansions. Like before, we will consider only continuous, finite functions defined on the interval $[-1,1]$. Let’s say we have a basis of functions that are orthogonal under the inner product

$$⟨f,g⟩={\int }_{-1}^{1}f(x)g(x)\mathrm{d}x.$$

One example of this is the trigonometric functions $\sin (k\pi x)$ and $\cos (k\pi x)$ for all non-negative integers $k$. It’s beyond the scope of undergraduate physics courses to prove that the trigonometric functions span this space, but they do, and you also can verify that the inner-product integral is indeed zero for any unequal pair. We’ll refer to elements of this basis as ${\varphi }_n(x)$, where $n$ is just a unique label.

Now, since the functions span the space of functions we can write down any function $f$ in terms of our basis ${\varphi }_n$ and some scalar coefficients $c_n$ as

$$f(x)=\sum _{n=0}^{\infty }{c}_n{\varphi }_n(x),$$

and this series is unique. For convenience, we’ll call the series representation $F_f$ to distinguish it from $f$ while we’re still determining the coefficients.

In the introduction to this series of articles, we introduced the idea of a series expansion, and saw the explicit forms of the Taylor and Fourier series. We are building towards a generalised series expansions based on “orthogonal” polynomials, but first we have to define some concepts from linear algebra. This article is roughly at the level of early first-year physics undergradutes.

Students at this level have come across the word “orthogonal” before when talking about Euclidean (“normal”) vectors. Here it means the same thing as “perpendicular” or “at right angles”, at least while you have three or fewer dimensions. Once you have more than that, or you’re dealing with some other type of vector, the definition is a little more abstract.

In all my time as a PhD student I have been involved with teaching undergraduate and postgraduate physics students at Imperial College London (they even gave me a prize in 2018!) as a teaching assistant in classroom-style tutorials and seminars. For my final year, though, I’ve also moved up into helping write the teaching materials, particularly for the differential equations part of the second-year course.

The first differential equations seminar we had was showing off some of the properties and uses of the Legendre polynomials, which the students had just met by solving Legendre’s equation. My part of the seminar was illustrating how any orthogonal basis of functions can be used to make a series expansion, and then getting the students to investigate how different types of functional expansion behave at different orders. If you just want to play with this, load up the Jupyter notebook on Colab or myBinder. The source code for this notebook is available on GitHub. You can see an example plot of these below.

The end result we’ll achieve mathematically is an abstract way of making series expansions. In general, a series expansion approximates a function $f(x)$ by using a (possibly infinite) sequence of terms, where each term is a constant multiplied by some basis function. Different series expansions use different bases and different methods of determining the coefficients.

Perhaps the most familiar example is the Taylor series. The Taylor series $T_f(x)$ that approximates a function $f(x)$ around some point $x_0$ is