Mathemagical Goodness

It's rare that I see a bit of maths and am just blown away. Tonight I stumbled upon Wallis product formula, or more specifically, its derivation and literally just thought "wow, that's amazing." Now, the Wallis product formula is something I have seen before, several times, but I just never paid it much attention. Now I see that it's so beautiful and so simple, no wonder it was worked out in 1655.

The basic idea is this. We know that the sine function has a well behaved Taylor series (it is an exponential function after all) so it makes sense to treat the sine function as the limit of a sequence of polynomials. The same logic works (as far as I can see) for any function with an absolutely convergent Taylor series.

Now polynomials have two basic representations. They can either be written as a sum of monomials like or they can be written as the product of factors like where the r's are the roots of the polynomial and A is just a (constant) scaling factor out front.

Now, assuming that the sine function has such a product representation the trick is to find A. We already know the r's: the roots of the sine function are simply the integer multiples of pi. So we expect thatand we simply need to find the constant A. To do this we simply need to evaluate both sides at a particular value of x. It could get difficult to do this though unless we pick a nice value for x. Let's try x=0. But both sides are zero when x is zero so this is no good! Solution: divide both sides by x and take the limit as x goes to zero. This is a common trick really, and it works a treat here.

Now we know A. All that remains is to put it together. But notice that there is one term in the denominator in A for each term in the product from sin(x)/x. We can take a typical term and simplify it like this . So the product ends up being:



So there you have the Wallis product formula for the sine function. From there it is easy to find a product formula for various multiples of pi etc by just plugging in values and rearranging to suite.

I realize that it is pretty straightforward maths and rather boring for most people probably. However I just found the beauty and simplicity of this product formula for such a common function worth sharing. I also realise that the LaTeX in this post is ugly and inconsistent but I have no good technique for rendering equations for this blog at the moment.

I hope that, with a touch of holiday spirit, you might forgive this silly maths nut. And maybe even find a little inspiration.