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.

Sure, Fourier series would be one way to look at it. A periodic function has a Fourier sequence (not series, since it's only a countably infinite number of Fourier coefficients) with zero coefficients everywhere BUT at a fundamental frequency f_0, and harmonics 2f_0, 3f_0, ... (It could be that there are some k for which kf_0 is zero, but all the nonzero coefficients HAVE to be at integer multiples of the fundamental.) if it doesn't behave that way, it's not periodic.

Without graphing it.... quickest way is: experience, and knowing the general shapes of "common" functions

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.

and elliptic functions are doubly periodic: http://mathworld.wolfram.com/EllipticFunction.html General info here: http://mathworld.wolfram.com/PeriodicFunction.html Edit: 2:30 A.M..... Back to studying math