Now I want to consider the differential equation . As I mentioned at the end of last time, we can write this as and find two solutions — and — by taking the two complex square roots of . But the equation doesn’t use any complex numbers. Surely we can find real-valued functions that work.
Indeed, we can, and we’ll use the same techniques as we did before. We again find that any solution must be infinitely differentiable, and so we will assume that it’s analytic. Thus we write
and we take the first two derivatives
The equation then reads
for every natural number . The values and are not specified, and we can use them to set initial conditions.
We pick two sets of initial conditions to focus on. In the first case, and , while in the second case and . We call these two solutions the “sine” and “cosine” functions, respectively, writing them as and .
Let’s work out the series for the cosine function. We start with , and the recurrence relation tells us that all the odd terms will be zero. So let’s just write out the even terms . First off, . Then to move from to we multiply by . So in moving from all the way to we’ve multiplied by times, and we’ve multiplied up every number from to . That is, we have , and we have the series
This isn’t the usual form for a power series, but it’s more compact than including all the odd terms. A similar line of reasoning leads to the following series expansion for the sine function:
Any other solution with and then can be written as .
In particular, consider the first solutions we found above: and . Each of them has , and , depending on which solution we pick. That is, we can write , and .
Of course, the second of these equations is just the complex conjugate of the first, and so it’s unsurprising. The first, however, is called “Euler’s formula”, because it was proved by Roger Cotes. It’s been seen as particularly miraculous, but this is mostly because people’s first exposure to the sine and cosine functions usually comes from a completely different route, and the relationship between exponentials and trigonometry seems utterly mysterious. Seen from the perspective of differential equations (and other viewpoints we’ll see sooner or later) it’s the most natural thing in the world.
Euler’s formula also lets us translate back from trigonometry into exponentials:
And from these formulæ and the differentiation rules for exponentials we can easily work out the differentiation rules for the sine and cosine functions: