 For whatever reason, the universe we live in is filled with periodic functions. We define the mass follows. Let c be greater than zero. If f of x plus c is equal to f of x for all x, then c is a period of f, and f is a periodic function. The least value c for which f of x plus c is equal to f for all x is the fundamental period. So let's consider a couple of familiar functions. f of x equals 3x, g of x equals 1, and h of x equals sine of x, and for the periodic functions, find the least period, if possible. So again, definitions are the whole of mathematics. All else is commentary. Let's pull in our definition of periodic function, and suppose there is a c for which f of x plus c is equal to f of x for all x, then and doing a little bit of algebra, we find that so this equality will only be true if c is equal to zero. But the requirement for a periodic function is that c is greater than zero, and so f cannot be periodic. What about g of x equals 1? Again, suppose there is a c for which g of x plus c is equal to g of x for all x, then, but this is true for any value of c, and so g is periodic, but there is no least period. How about for sine of x? Again, suppose there is a c for which h of x plus c is equal to h of x for all x, then, and this is true for c equal to pi n, where n is any whole number, and so sine of x is a periodic function, and the least c for which this is true is c equal to 2 pi, and so our fundamental period is 2 pi. And you might remember from trigonometry a more general result, the period of sine of kx, or cosine of kx, is 2 pi over k. No function is an island, and all functions work better when they work together. Suppose f and g are periodic functions, then we have the following useful theorem. If f and g are periodic, any linear combination of f and g will also be periodic. And this is something that you should be able to prove. So let's show that f of x equals sine of 3x plus cosine of 5x is periodic and determine its fundamental period. So it's useful to remember our theorem that the period of sine kx or cosine kx is 2 pi over k. Since sine of 3x has period 2 pi over 3, and cosine 5x has period 2 pi over 5, then both functions will have the same value when m periods of sine of 3x equals n periods of cosine 5x. And so we want 2 pi over 3m to be equal to 2 pi over 5n. We'll do a little algebra, and we note that the least value for which this will occur is m equal to 3n equal to 5. In other words, 3 periods of sine of 3x will have the same duration as 5 periods of sine of 5x. And 3 periods of sine of 3x will be an interval of length 2 pi. And we might summarize our results here. We note that sine of 3x plus 2 pi is equal to, which is the same as sine of 3x. Likewise, cosine of 5 times x plus 2 pi is cosine of 2x. And so f of x plus 2 pi is equal to f of x. So f is periodic. Moreover, remember this is the least value for which this will occur. And so 2 pi is the smallest c for which f of x plus c is equal to f of x. And so the fundamental period is 2 pi. Now one of the things that makes periodic functions so useful is that if f is periodic, then since f of x plus c is equal to f of x, we only need to specify f over 1 period between a and a plus c. And since cosine and sine are the most familiar periodic functions, we might ask the following question. If f is a periodic function, could we express f as a series in cosine and sine where our index n is in some set of real numbers? So let's think about that. Suppose our function has fundamental period l. We might begin by looking for other functions with period l. Suppose we want sine of kx to have period l. Remember the period of sine of kx or cosine of kx is 2 pi over k. So we require 2 pi over k to be l. And so that means that k has to be 2 pi over l. And the same argument holds for cosine of kx. And so we note that sine of 2 pi over lx and cosine of 2 pi over lx have period l. But wait, there's more. While f has fundamental period l, we only care about other functions with period l. And so remember, any multiple of a fundamental period is a period. And so while these two functions have period l, so does sine of 2 pi n over lx and cosine of 2 pi n over lx for any integer n. And this leads to the following idea. Take l not equal to zero, then g of x, the infinite series whose terms are an cosine 2 pi n over lx plus bn sine 2 pi n over lx has period l. Provided we have convergence, remember any results regarding an infinite series are meaningless without convergence. Conversely, this suggests that if f has fundamental period l, it might be expressible in the form of a trigonometric series. In 1807, a French mathematician began the study of these series in an effort to solve a partial differential equation. We'll talk about that one later. The mathematician's name is spelled this way, and should probably be pronounced foyer, but English speakers have a hard time with this. And so we often pronounce this, no, no foyer series. Now this convergence question is actually very difficult, and even today there are questions regarding the convergence of Fourier series. So for the most part at this level, we're not going to worry about convergence, but we do have the question, how can we find an and bn? We'll take a look at that next.