Taylor series: an indispensible tool
Sometimes we find ourselves dealing with a function, let’s call it g(x), which is ‘hard’ to work with (…perhaps because it has some rather complicated structure); at the same time, we may find that we are only interested in some small range of values of x. In these circumstances life becomes much simpler if we can find a simple approximation to g(x), that is ‘good’ in the range of interest. The motion of a pendulum provides a simple example. The restoring force on the pendulum varies as sinθ, where θ is the angular displacement. The sine function is actually quite ‘hard’ to handle here. However if we are concerned only with what goes on for small θ (…the motion is small in amplitude) we can use the approximation sinθ≈θ, which, you probably know, is ‘good’ for ‘small θ’. This is one instance of the Taylor expansion – possibly the most widely used approximation tool in the physicist’s toolkit.
It is quite likely that you will not have met this bit of mathematics before. We won’t provide you with a complete proof here …your mathematics courses will see to that sooner or later We will just write down the basic result and dig the sense out of it. To make things look as simple as possible we will suppose that the point of particular interest is x=0. There is no real loss of generality here: we can always imagine choosing our ‘x-axis’ so that this is true.
The claim is that, for small enough x, we can approximate the function g(x) by an expansion in powers of x:
![\[ g(x) = g(0) + g'(0)x + g''(0)\frac{x^{2}}{2!} + g'''(0)\frac{x^{3}}{3!} + \ldots \]](mastermathpng-0.png)
In this expression,
![\[ g'(0) \equiv \frac{dg(x)}{dx}\mid_{x=0} \]](mastermathpng-1.png)
To see how this expansion comes about ask yourself this question: what is the simplest possible way to approximate the behaviour of a function g(x) in the neighbourhood of the point x=0? Answer — as a constant. What constant? Answer — the value of the function at x=0. Obvious, isn’t it? There you have the first term of the Maclaurin series g(x)=g(0)+… (the dotted line in the figure). Second question: under what conditions will this be the best approximation? Answer: when the slope of the curve at x=0 is zero, i.e. the curve is ’flat’ at the origin. What if the curve is not flat, i.e. its tangent at the origin is a slanting straight line? Well, then we can add a term so as to approximate the curve by just this straight line. It passes through the point g(0), and has slope equal to that of the tangent, i.e.g'(0). So, g(x)=g(0)+g'(0)x (…remember y=mx+c ?); a second term in our DIY expansion (the dashed-dotted line in the figure). Perhaps you can now make sense of the third term (the dashed line)?
See Q7.2 for some practice with the Taylor Series expansion. And be fore-warned: you are going to meet it frequently if you proceed to further courses in physics.