0

how to determine if the following function is periodic, and if so how to determine its period. \begin{equation} \sum_{n=1}^{\infty} \cos \frac{n \pi x}{l} .....(1) \end{equation}

I know that the function f(x) is periodic if f(x+T)=f(x) for every x; where T is the period of the function f(x). I can solve that the period of \begin{equation} \cos \frac{n \pi x}{l} \end{equation} is \begin{equation} \frac{2 l}{n} \end{equation} but don't understand the problem (1) above.

I'm studying fourier series

wait, latext

amer hakam
  • 1

2 Answers2

0

This statement you made is not the full story:

the function $f(x)$ is periodic if $f(x+T)=f(x)$ for every $x$; where $T$ is the period of the function $f(x)$.

The full story is:

the function $f(x)$ is periodic if $f(x+T)=f(x)$ for every $x$; where $T$ is a period of the function $f(x)$.

Periodic functions actually have infinitely many periods. If $T$ is a period of $f$, then $$f(x + 2T) = f((x + T) + T) = f(x + T) = f(x)$$ So $2T$ is also a period of $f$. And an almost identical calculation shows that $3T$ is also a period, and so on. If $T$ is a period of $f$, then so is $kT$ for any positive integer $k$.

Now if $f$ is continuous and periodic but not constant, there has to be a smallest period, and that is the one we normally call the period of $f$. But it is useful to keep in mind that all the multiples of that value are also periods.

Next note that if two functions $f$ and $g$ both share a common period $T$ then so does any operation applied to them. If $F(x,y)$ is any function of two variables, then $$F(f(x+T),g(x+T))= F(f(x), g(x))$$

The same holds true for any number of arguments, even infinite, provided the individual functions are all periodic with the same period $T$. $$F(f_1(x+T), f_2(x + T), f_3(x + T), \dots) = F(f_1(x), f_2(x), f_3(x), \dots)$$

The summation in your problem is a function $F$ of infinitely many variables. So if you could find a common period for all of the $\cos \frac{n\pi x}l$ functions being added, it would also be a period for their sum.

Paul Sinclair
  • 43,643
  • This should be a comment, not an answer. – Will Sherwood Aug 31 '22 at 22:05
  • @WillSherwood - Either you are joking, or you haven't understood at all my point. If the latter, then I must bear some responsibility for having failed to be clear enough. But even so, this is an answer, not a comment - whether you can recognize it as such or not. It is true I have not simply written out the answer, but I practically rubbed your nose in it. – Paul Sinclair Sep 01 '22 at 01:22
  • I'm not joking - your answer is not specific enough to the class of functions listed in the question, and it is not general enough to solve the problem over all classes... – Will Sherwood Sep 01 '22 at 01:36
  • What you wrote basically amounts to "To find the period of a sum of periodic functions, find the common period amongst them!" – Will Sherwood Sep 01 '22 at 01:37
  • Plus a bunch of facts about periodic functions, which are not only trivial in hindsight, but I would actually say are just straight up trivial.. – Will Sherwood Sep 01 '22 at 01:43
0

You are summing functions of period $2\ell$, $\ell$, $2\ell/3$, $\ell/2$, $2\ell/5$, and so on. Notice that each of these evenly divide $2\ell$. Hence, the period of $f$ is $2\ell$.

Will Sherwood
  • 1,371
  • Note that as originally stated, the sum actually diverges. But if you divide each summand by $n$, you will get the desired result. – Will Sherwood Aug 31 '22 at 22:07
  • So, as written, your $f$ is not a function and hence not periodic. If you write $f = \sum_{n=1}^\infty \frac 1 n \cos\left(\frac{n\pi x}{\ell}\right)$, then the periods of the summands do not change, but now $f$ converges, so the period of $f$ is $2\ell$. – Will Sherwood Aug 31 '22 at 22:09