User:Kieff/Iterated trigonometric functions
From Wikipedia, the free encyclopedia
Iterated trigonometric functions are functions built by using trigonometric functions recursively. The first useful one I stumbled upon was the iterated sine function, which I defined as follows:
- sin0x = x
- sin1x = sin(x)
- sinix = sin(sini − 1(x))
This function is interesting in that it arbitrarily approaches a continuous and smooth square wave, without any ringing artifacts, if you append to it a normalization factor as i tends to infinity (this is important otherwise it'll converge to zero).
I was wondering if I could approach other primitive waveforms (such as triangle and sawtooth waves) with a similar method. I found that tan(sin(x)) is close to a smooth triangle wave, but I couldn't manage to make it arbitrarily close. sin(tan(x)) looks like a sawtooth wave, but it gets nasty as the tangent goes to infinity.
Interestingly, iterating tan(sin(x)) 7 times gets pretty close of the square sine function, but it still doesn't get arbitrarily close to it.
On the other hand, iterating cosine and tangent alone gave uninteresting results.
