What is the average value of a bounded periodic function over a period?

[kent davidge]

Does it make sense to just talk about the average value of a function without specifying the range over which the average is taken? It seems a common occurrence in discussions of waves to just mention that the average value of the complex exponential $e^{ix}$ is zero. But it will be zero only if we look at it over a $2\pi$ interval, like from $-\pi$ to $\pi$, correct?

 

[mathman]

Your asertion is correct. However in many instances, the domain is obvious, so it is not stated explicitly, such as one period for sine waves, etc.

 

[HallsofIvy]

If a function is periodic then it makes sense to refer to the average over one period as the average of the function.

 

[zinq]

Another way to view the average of a bounded periodic function is to take the limit of its average over an increasingly large domain. So that if, say, f : R → R is a bounded continuous function with some period p > 0, so that f(x + p) = f(x) for all x, then the limit as T → ∞ of 1/(2T) times the integral of f(x) over the interval [-T, T] will approach the same value as its average over one full period.

So if we know f is a bounded continuous periodic function, we can define its average over a period without knowing what that period is.