Approximating even functions by powers of cosine

A couple days ago I wrote a post about turning a trick into a technique, finding another use for a clever way to construct simple, accurate approximations. I used as my example approximating the Bessel function J(x) with (1 + cos(x))/2. I learned via a helpful comment on Mathstodon that my approximation was the first-order part of a more general series

$J_0(x) = 1 + \frac{\cos(x) - 1}{2} - \frac{(\cos(x) - 1)^2}{48} + \frac{7(\cos(x) - 1)^3}{1440} + \cdots$

The first-order approximation has error O(x⁴), as shown in the earlier post. Adding the second-order term makes the error O(x⁶), and adding the third-order term makes it O(x⁸).

I’ve written a few times about cosine approximations to the normal probability density. For example, see this post. We could use the same idea as the series above to approximate the normal density with a series of powers of cosine. This gives us

$\exp(-x^2/2) = 1 + (\cos(x) - 1) + \frac{(\cos(x) - 1)^2}{3} + \frac{2(\cos(x) - 1)^3}{45} + \cdots$

and as before, the first, second, and third order truncated series have error O(x⁴), O(x⁶), and O(x⁸).

The general theory behind what’s going on here is an extension of Bürmann’s theorem. The original version of the theorem relies on a series inversion theorem that in turn relies on the approximating function, in our case cos(x) − 1, not having zero derivative at the center of the series. But there is a more general form of Bürmann’s theorem based on a more general form of series inversion. We will always need a more general version of the theorem when working with even functions because even functions have zero derivative at zero.

Here’s another example, this time using the Bessel function J₁, an odd function, which does use the original version of Bürmann’s theorem to approximate J₁ by powers of sine.

$J_1(x) = \frac{1}{2} \sin(x) + \frac{1}{48} \sin^3(x) + \frac{17}{1920} \sin^5(x) + \cdots$

In this case truncating the series after sin^k(x) gives an error O(x^{k + 2}).

You can find more on Bürmann’s theorem in Whittker and Watson.

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