1

What is the fundamental period of the function $$ f(x) = \sin x + \tan x + \tan\frac{x}{2} + \tan\frac{x}{4} + \tan\frac{x}{8} + \tan\frac{x}{16}~ .$$

I know that $16\pi$ is one period but how can I determine the fundamental period?

Can anyone please help me to find out it's fundamental period?

My friend was telling me that it's fundamental period will also be $~16\pi~$. Because $~16\pi~$ is the L.C.M of all periods of the periodic functions in the expression.

But I can not understand this argument because the well known function $~|\sin x | + |\cos x|~$ is a periodic function with period $~\frac{\pi}{2}~$ where as $~|\sin x |~$ and $~|\cos x|~$ are of period $~\pi ~$.

nmasanta
  • 9,222
cmi
  • 3,371
  • Last term will change to $\cot \frac{x}{16}$ @Jean-ClaudeArbaut – cmi Aug 14 '19 at 06:15
  • Then what is happening? – cmi Aug 14 '19 at 06:20
  • Ohh sorry.. I did not see properly.. It will be $\infty$ which is different from $f(0)$ .So $8\pi$ Can not be fundamental period.Thus similarly for any number in your list which is less than $16\pi$ can not be the fundamental period.. Is this what you are trying to say?@Jean-ClaudeArbaut – cmi Aug 14 '19 at 06:35
  • Yes. ${}{}{}{}{}$ – Jean-Claude Arbaut Aug 14 '19 at 06:47
  • So you are saying if $T$ is the period then fundamental period will definitely be a member of [$\frac{T}{n}$:$n \in \mathbb N$ ]? @Jean-ClaudeArbaut – cmi Aug 14 '19 at 06:58
  • Posted twice within an hour. Why?https://math.stackexchange.com/questions/3322860/the-fundamental-period-of-the-function-fx – Gerry Myerson Aug 14 '19 at 07:33
  • 1
    I am having a network issue.. I was trying to edit the post..But it asked to change the title. Then I saw that a new post had been posted already.. I was trying to delete that.. But I could not..@GerryMyerson – cmi Aug 14 '19 at 07:38
  • Yes. If $T_f$ is the fundamental period (and if it exists!), then any period $T$ is an integer multiple of $T_f$. – Jean-Claude Arbaut Aug 14 '19 at 15:28

3 Answers3

3

It is easy to check that $f(x + 16\pi) = f(x)$ for all $x \in \mathbb{R}$. Since $f$ is continuous on its domain (or even continuous as function $\mathbb{R} \to \mathbb{R}\cup\{\infty\}$) and non-constant, it follows that the fundamental period of $f$ is of the form $16\pi/n$ for some positive integer $n$.

Now if $n$ is a positive integer for which $16\pi/n$ is a period of $f$, then we must have $f(16\pi/n) = 0$.

  • If $n > 32$, then it is easy to see that each summand of $f(16\pi/n)$ is positive, and so, $f(16\pi/n) > 0$.

  • So it suffices to check that $f(16\pi/n) \neq 0$ for each $2 \leq n \leq 32$. This is the trickiest part, and to be honest, I do not see any clear argument for this. (Although we can easily remove $n = 2, 4, 8, 16, 32$ out of options, all the other values still deserve to be investigated.) But any CAS is capable of computing those values, and it turns that none of them are zero.

  • Therefore the only possible choice is $16\pi$.

Sangchul Lee
  • 167,468
  • So continuity is the main thing. If the function was not continuous then , we could not have said that the fundamental period will belong to [$\frac{16\pi}{n}$:$n \in \mathbb N$ ]..Am i right?@Sangchul Lee – cmi Aug 14 '19 at 07:05
  • @cmi, That is right. Continuity is one such condition that prevents from having 'infinitesimal periods' (or more precisely, arbitrarily small periods), thus enabling the notion of 'fundamental period'. – Sangchul Lee Aug 14 '19 at 07:08
  • Yea that is what I was missing..Can you please tell me how continuity would make this possible..@Sangchul Lee – cmi Aug 14 '19 at 07:25
  • @cmi, For any function $f$ on $\mathbb{R}$, it is easy to check that the set of all periods $$P={a\in\mathbb{R}:f(x+a)=f(x)\text{ for all }x\in\mathbb{R}}$$ forms an additive subgroup of $\mathbb{R}$. Then as we can confirm from this posting, we have a trichotomy: (1) $G={0}$, (2) $G$ is dense in $\mathbb{R}$, or (3) $G$ is of the form $G=T\mathbb{Z}$ for some $T>0$. But if $f$ is periodic, non-constant and continuous, then neither of (1) nor (2) occurs, and so, only (3) is possible. Then of course, $T$ is the fundamental period of $f$. – Sangchul Lee Aug 14 '19 at 07:31
  • By the way this is not a continuos function...Can you please elaborate little bit more? – cmi Aug 14 '19 at 07:31
  • By the way this is not a continuos function...Can you please elaborate little bit more? @Sangchul Lee – cmi Aug 14 '19 at 07:40
  • @cmi, As I alluded in my answer, we may regard $f$ as a function from $\mathbb{R}$ to the one-point compactification $\mathbb{R}\cup{\infty}$ of the real line. This is topologically the same as the unit circle via the stereographic projection. Under this consideration, we can prove that $f$ is continuous and all the aforementioned discussions still work. Of course, without this fancy machinery we can still justify the existence of fundamental period with due modification, although the proof will be less elegant. – Sangchul Lee Aug 14 '19 at 07:52
  • I got it..Thank you for patiently hearing my doubts and clear them..@Sangchul Lee – cmi Aug 14 '19 at 07:55
  • Glad it helped. :) My answer is still not satisfactory as it leaves the most crucial part to calculators. One possible approach that I am suspecting is to write $f(x)=R(e^{ix/8})$ for a rational function $R$ and then investigate the property of $R$. – Sangchul Lee Aug 14 '19 at 08:09
0

In the given function 16π is the fundamental period. The reason is if "T " is one of period then $ \frac{T}{n} $ where n$ {\in N } $ can be a fundamental period if $\\ f(x\,+\,T)\,=f(x) $ . Generally it happens in function which transform from one to the other given e.g |Sin((π/2)+x)| = | Cos x | and vise versa there fundamental period get refused to π/2.

mathsdiscussion.com
  • 1,235
  • Do you know Dirichilet function? Then what is the fundamental period of dirichilet function by your theorem?@mathdiscussion.com – cmi Aug 14 '19 at 06:44
  • Remember that I am talking with respect to the given problem only other wise there is only one definition of periodic function if there exist any T>0 such that f(x+T)=f(x) then f(x) periodic and least such positive T is fundamental period. This hold for all functions except constant function in which case any real number is one of its period whereas fundamental period is not well defined. – mathsdiscussion.com Aug 14 '19 at 07:17
0

If $T_1,T_2,T_3,...$ are the periods of individually added functions and if the ratios of these periods are rational the period of the the whole function is given by the $LCM(T_1,T_2,T_3...)$ If $T$ is period of $g(x)$, the period of $g(ax)$ is $T/a$. So the period of the whole given function is $$T=LCM(2\pi,\pi,2\pi, 4\pi, 8\pi, 16 \pi)=16 \pi$$

Z Ahmed
  • 43,235