Finding period of any type of function

[Raghav Gupta]

Homework Statement

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?

Homework 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

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?

 

[HallsofIvy]

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

 

[SammyS]

Homework Statement

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

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

Homework 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

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?

Try graphing y = x - [x] .

 

[Raghav Gupta]

Try graphing y = x - [x] .

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?

 

[Mark44]

Graphing fast seems difficult without a graph paper and scale.

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

 

[Raghav Gupta]

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

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.

 

[haruspex]

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

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

 

[Raghav Gupta]

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

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.

 

[haruspex]

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.

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

 

[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?

 

[haruspex]

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

They canceled 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?

 

[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}

 

[haruspex]

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}

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.

 

[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 ?

 

[haruspex]

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 ?

Any integer.

 

[Raghav Gupta]

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.

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

 

[haruspex]

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

Conjecture, I wrote.

 

[Raghav Gupta]

Conjecture, I wrote.

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. ?

 

[RUber]

You seem to have missed the part about coprimes.

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.

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.

 

[Raghav Gupta]

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

 

[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?

 

[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.

 

[Raghav Gupta]

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.

Not understanding that bold part.

 

[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.

 

[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.

 

[Raghav Gupta]

I was doing some manipulations and got
$sin^4x + cos^4x = 1-2sin^2xcos^2x = 1- 1/2 (sin^22x)$
Now period of sin 2x is π. Hence square of it would have π/2.

 

[Raghav Gupta]

@haruspex and @RUber why the conjecture abx/n is not applicable in the last two examples for finding period of them?

 

[haruspex]

@haruspex and @RUber why the conjecture abx/n is not applicable in the last two examples for finding period of them?

How does it break the conjecture? Put a=b=1, n=2, x=pi.

 

[Raghav Gupta]

Why we are choosing here n= 2? 1,1 are co-prime and putting n= 1 we then get not the desired result.
If n and ab must be co prime the value of n can be any integer in this case?

 

[RUber]

The conjecture that @haruspex stated earlier was that the LCM always works, i.e. n=1 will give you a period, but there may be a smaller period due to cancellations or other properties of the function.
sin x * sin x is periodic over 2pi, but it is also periodic over pi. How and when you can use n>1 is not discussed in the argument, and I imagine for more complicated functions might really require a good look under the hood to understand.
However, the bottom line is the LCM is always safe. With analysis of the function, you might be able to find an n>1 to reduce the period for certain combinations.