Originally posted by: ga14
How do you determine if a function is periodic or not without graphing it?
For example, cos^2(2pit) is periodic, as is sin^3(2t). But e^(-2t)cos(2pit) is nonperiodic, as is the discrete signal x[n]=cos(2n).
Does this have to do with Fourier series somehow? Is there an easier way to determine it? Thanks for any help.
I just came up with a little something that you might find interesting. I thought about it while I was working on some math (before my quiz tomorrow).
In the case y = a sin b (where a and b are functions
of x, not just constants), the function y is periodic if the derivative of both of the functions is equal to zero and a constant, respectively. In your aforementioned example, the derivative of 2pit with respect to t would indeed be a constant (2pi), but, using the chain rule, the derivative of e^(-2t) is -2e^(-2t), which has obvious fluctuations, no matter how many times you take the derivative. (In fact, each time you take the derivative, you would be messing up your function by a factor of 2 each time.)
...so that's one way to tell if a trigonometric function is periodic, I suppose.