Periodic Functions

Sine(x) - the most common periodic function with period 2
[Tangent Graph]

Tangent(x) - another periodic function with period

Overview

Periodic functions are functions that repeat over and over, or cycle on a specific period. This is expressed mathematically that

A function is periodic if "there exists some number p>0 such that f(x)=f(x+p) for all possible values of x" [1.7,p.112]

The fundamental period of a function is the length of a smallest continuous portion of the domain over which the function completes a cycle. That is, it's the smallest length of domain that if you took the function over that length and made an infinite number of copies of it, and laid them end to end, you would have the original function.
[Sine graph]

For example, let's say that we have some imaginary function f(x) and that f(0.1)=2.35 if f(x) is a periodic function with period 2, then f(2.1)=2.35, because 2.1 is exactly 0.1+2. That is, after 2 units, you're back where you started. This continues up and down the number line because the function is periodic, so:
f(4.1)=f(6.1)=f(8.1)=f(-1.9)=f(0.1)=2.35.
We can generalize this by saying that for our function f(x), f(2*k+0.1)=2.35, k integer. To further generalize the pattern, we can define an arbitrary periodic function g(x) with period p by saying that:
f(p*k+a) = f(a) is true for all a real, k integer.

A property of some periodic funtions that cycle within some definite range is that they have an amplitude in addition to a period. The amplitude of a periodic function is the distance between the highest point and the lowest point, divided by two. For example, sin(x) and cos(x) have amplitudes of 1.

Combining Periodic Functions

Because sine and cosine are both periodic and have the same period, when you add them up, subtract them, multiply them, etc you get functions that are also periodic. This can be proven with some care - PROOF - Let us assume that:
f(p*k+a) = f(a) is true for all a real, k integer.
Simply multiplying each side by some constant does not change the equation, nor does adding or subtracting some constant to each side change the periodicity:
(substitute constant K = f(p*k+a) = f(a) )
c*K = c*K K + c = K + c
But this is simple equation manipulation. What about when we mix different functions? If we have two functions, f(x) & g(x), with the same period, we can throw them together any way we want because, if their period is p, and at any value a:

Homework Assignment: Extend this proof for multiplication - use h(x) = f(x)*g(x) f(a) = c1; g(a) = c2, so f(a) + g(a) = c1 + c2 = C f(a) + g(a) = C
Also, because they are periodic, for k integer:
f(a) = f(p*k+a) = c1 g(a) = g(p*k+a) = c2 f(p*k+a) + g(p*k+a) = c1 + c2 = C
To make things clearer, many people use an auxilury function h(x):
h(x) = f(x) + g(x) h(a) = f(x) + g(x) = C h(p*k+a) = f(p*k+a) + g(p*k+a) = C Thus, h(a) = h(p*k+a)

What about functions that don't have the same fundamental period? Well, that's a different story. Let's say we have two (new) periodic functions f(x) and g(x), with periods p and r, respectively:
f(p*k+a) = f(a) is true for all a real, k integer.
g(r*k+a) = g(a) is true for all a real, k integer.
Now we can't construct that nice h(x) like we did because we have clashing periods. (Not to be mistaken with cat fights resulting from two women with PMS at the same time) So what can we do? Functions don't have just one period! If a function repeats every 2 units, then it will also repeat every 6 units, won't it? So if we have one function with fundamental period 2, and another with fundamental period 8, we've got no problem because 8 is a multiple of 2, and both functions will cycle every 8 units.
So, if we can "patch up" the periods to be the same, we know that if we combine them, we'll get a function with the "patched up" period.
What about if f(x) has period 3 and g(x) has period 4? We can easily see that in 12 units, both will cycle, so they're fine. What about fractions? Let's say that we had functions with periods 13/12 - f(x) and 2/21 - g(x). What we need are two numbers of periods that we can multiply by the periods to get some common, "patched-up period". First, we can simplify the problem by multiplying each period by its denominator to find whole number periods. So we know that f(x) has a period of 13 (in 12 fundamental periods) and g(x) a period of 2 (in 21 fundamental periods). Now we can simply do what we did in the 3 and 4 case and multiply by both periods to find a period for the new combination function. So h(x) has a period of 26. There's a walk through pseudo-proof for the following Theorem (generalize the proof for homework!)

If two periodic functions, f(x) and g(x), have rational periods, then any addition or multiplication combination of the functions (not composition!) will also be periodic.

Now composition of different irrational numbers, like e and , is a different story. Someone from number theory - a proof would be greatly appreciated.

Also, for those who like digging, I challenge you to prove or disprove that if f(x) is a periodic function and g(x) is not a periodic function, then g(f(x)) is periodic and f(g(x)) is not.
-David