Homework Help: Finding period of any type of function

1. Mar 25, 2015

Raghav Gupta

1. The problem statement, all variables and given/known data

1. The period of the function f(x) = tanπx + x - [x], where [.] is the greatest integer or floor function is?

2. The period of the function f(x) = sin2πx + x - [x], where [.] is the greatest integer or floor function is?

2. Relevant equations
For sin (ax+ b) + c where a,b,c are constants the period is 2π/a
For tan(ax+ b) + c where a,b,c are constants the period is π/a

3. The attempt at a solution
In first problem one can do π/π and say period of it is one. But the terms x - [x] has also to be looked.
How also to say that it is periodic?
I know the definition for periodicity,
ƒ(x + T) = f(x) for any x, where T is time period.
But how to check it in this complex seeming equations?

Last edited: Mar 25, 2015
2. Mar 25, 2015

HallsofIvy

The period of the sum of two periodic functions is the least common multiple of the two periods.

Last edited by a moderator: Mar 25, 2015
3. Mar 25, 2015

SammyS

Staff Emeritus
Try graphing y = x - [x] .

4. Mar 25, 2015

Raghav Gupta

Graphing fast seems difficult without a graph paper and scale.

I know that for y= x2 it's a upward parabola
For y= x it's a straight line.

In general for any equation graphing what's the best way to graph it?

5. Mar 25, 2015

Staff: Mentor

You can do a quick sketch without needing graph paper, and a ruler isn't an absolute essential.

6. Mar 25, 2015

Raghav Gupta

Yeah, got the graph and period of it as 1.
Thanks to you all.
Got the answer finally one in both the above mentioned problems.

Last edited: Mar 25, 2015
7. Mar 25, 2015

haruspex

Not necessarily - there could be some cancellation. E.g. add sin(x/2)+sin(x/3) and sin(x/5)-sin(x/3),

8. Mar 25, 2015

Raghav Gupta

But when we know that sinx/3 terms would cancel then why looking for the individuals period and then it's least multiple?
We would have sinx/2 and sinx/5 having individuals period 4 pi and 10 pi and their LCM 20 pi.

9. Mar 25, 2015

haruspex

Quite so. I was merely pointing out that Hall's formula is not always true.

10. Mar 25, 2015

Raghav Gupta

Here the terms were canceling. Do you know some other example where the terms are not cancelling like this and period is coming different?

11. Mar 25, 2015

haruspex

They cancelled because I constructed them that way. In general, given two periodic functions f and g, you might not have a representation of them as a sum of simple closed forms. E.g. they could be defined by indefinite integrals that cannot be solved. What would it mean to say that they have cancelling terms?
Perhaps one could define that if the period of f+g is less than the LCM of the individual periods then some cancellation has occurred, but I don't see that such a definition is of any interest.
What would a general rule be?

12. Mar 25, 2015

Raghav Gupta

I think we should take a example to verify this.
Suppose we have sin2πx + cos2πx then their individual periods are 1 and 1.
Now let's take the prime 2 which divides these m= 0 and n=0.
Now here f+ g has period 1.
Now the p=2 divides it 0 times between |0-0| and max{0,0}

13. Mar 26, 2015

haruspex

I see you read the last part of my post before I deleted it. It occurred to me that the period as an integer was not defined - periods are x-intervals. All we can talk about is rational ratios between periods.
In my example in post #7, all you would know (from observation) is that the periods are in the ratio 2:5, 2x and 5x, say. So I replace my previous conjecture with : if the periods of f and g are ax, bx, where a and b are coprime integers, then the period of f+g is abx/n where n and ab are coprime.

14. Mar 26, 2015

Raghav Gupta

I am having a bit difficulty in this. The proof of finding periods and a general statement making can be difficult.
What is n here in abx/n ?

15. Mar 26, 2015

haruspex

Any integer.

16. Mar 27, 2015

Raghav Gupta

What's the proof for this?
Hall's statement was intuitive a bit.

17. Mar 27, 2015

haruspex

Conjecture, I wrote.

18. Mar 27, 2015

Raghav Gupta

Oh I didn't know the meaning of conjecture before. I had seen this word used many times in Math but never looked its definition. Now I know.
So how to apply your conjecture here in the case of sinx/2 + sinx/5?
Here the period of f is 4π and of g is 10π.
By your conjecture period of f+ g here is,
4π*10π/n. = 40π2/n. ?

19. Mar 27, 2015

RUber

You seem to have missed the part about coprimes.
You should be looking for LCM of the periods, or use x in ax, bx, as the greatest common factor of the periods. It seems like @haruspex is saying that the LCM is the upper bound, but with the potential of cancellation, there may be an n, not equal to one, which could cut the period down.
A period function with period (a) is also periodic over intervals of (2a), (3a), (ka), etc. So you know for a fact, that the function will surely be periodic over the LCM of the periods. It may, however, have a smaller essential period.

20. Mar 28, 2015

Raghav Gupta

I am having a difficulty. No problem, would understand it later if required.

21. Mar 29, 2015

Raghav Gupta

I have found a perfect example to counter Hall's statement, in agreement with Haruspex and ruber.
|sinx| has period of π,
|cosx| also has period π,
But |sinx| + |cosx| has period π/2.
Can anyone tell me how?

22. Mar 29, 2015

RUber

Note that $|\sin x | = \left\{ \begin{array}{l l} \sin x & 0\leq x < \pi (mod 2\pi) \\ - \sin x & \pi\leq x < 2\pi (mod 2\pi) \end{array} \right.$
$|\cos x | =|\sin (x+\pi/2) | \left\{ \begin{array}{l l} \sin (x+\pi/2) & 0\leq x+\pi/2 < \pi (mod 2\pi) \\ - \sin (x+\pi/2) & \pi\leq x+\pi/2 < 2\pi (mod 2\pi) \end{array} \right.$
So, for example, for $x \in (0,\pi/2), |\sin x |+ |\cos x | = \sin x + \sin (x+pi/2)$
$x \in (\pi/2,\pi), |\sin x |+ |\cos x | = \sin x - \sin (x+pi/2)$
but this is the same as writing
$y \in (0, \pi/2), \sin (y+\pi/2) - \sin (y+pi)$
and $\sin (y+ \pi ) = - \sin y$
so, you have the same thing.

23. Mar 29, 2015

Raghav Gupta

Not understanding that bold part.

24. Mar 29, 2015

RUber

If this function has a period of pi/2, then f(x) = f(x+pi/2).
For x in (0,pi/2) then,
f(x) = sin(x) + sin(x+pi/2), right?
f(x+pi/2) = sin(x+pi/2) - sin(x+pi), right? Note the sign change since (x+pi) > pi.
Now, since sin(x+pi) = -sin(x), you can make the substitution to get:
f(x+pi/2) = sin(x+pi/2) - (- sin(x) ), which can be rearranged to get f(x+pi/2) = sin(x) + sin(x+pi/2).
Note that f(x) = f(x+pi/2). So, the period of f is pi/2.

25. Mar 30, 2015

Raghav Gupta

Got it Ruber thanks.
So when in general can we apply LCM rule?
Here it was modulus, so we had to be careful for + and - signs.
A last example $sin^4x + cos^4x$ also has period π/2 instead of π.