Proving a function is peridoic

  • Thread starter trap101
  • Start date
  • Tags
    Function
In summary, to prove that cos(x) + cos(\alphax) is periodic if \alpha is a rational number, we can use the definition of periodicity and solve for p by setting x+p equal to multiples of 2\pi and 2m\pi/\alpha for integers n and m. This is based on the fact that \cos x has period 2\pi and \cos(\alpha x) has period (2\pi/\alpha).
  • #1
trap101
342
0
Show that cos(x) + cos([itex]\alpha[/itex]x) is periodic if [itex]\alpha[/itex] is a rational number.


Ok So I don't think I have ever done a question proving periodicity. But by definition a function f is periodic if:

f(x + p) = f(x)

so then:

f(x+p) = cos(x +p) + cos([itex]\alpha[/itex](x+p))
= [cos(x)cos(p)-sin(x)sin(p)] + [cos([itex]\alpha[/itex]x)cos([itex]\alpha[/itex]p)-sin([itex]\alpha[/itex]x)sin([itex]\alpha[/itex]p)


and this is where I am stuck, so from what I've learned I have to solve for p is some fashion, I'm just not sure how,
 
Physics news on Phys.org
  • #2
hi trap101! :smile:

hint: cos(0) + cos([itex]\alpha[/itex]0) = 2 :wink:
 
  • #3
trap101 said:
Show that cos(x) + cos([itex]\alpha[/itex]x) is periodic if [itex]\alpha[/itex] is a rational number.

The first thing to do with a rational number is write it as [itex]\alpha = r/s[/itex] for co-prime integers r and s.

Ok So I don't think I have ever done a question proving periodicity. But by definition a function f is periodic if:

f(x + p) = f(x)

so then:

f(x+p) = cos(x +p) + cos([itex]\alpha[/itex](x+p))
= [cos(x)cos(p)-sin(x)sin(p)] + [cos([itex]\alpha[/itex]x)cos([itex]\alpha[/itex]p)-sin([itex]\alpha[/itex]x)sin([itex]\alpha[/itex]p)


and this is where I am stuck, so from what I've learned I have to solve for p is some fashion, I'm just not sure how,

You already know that [itex]\cos x[/itex] has period [itex]2\pi[/itex] and [itex]\cos(\alpha x)[/itex] has period [itex](2\pi/\alpha)[/itex]. So you need to find [itex]p[/itex] such that [itex]x + p = x + 2n\pi[/itex] and [itex]x + p = x + 2m\pi/\alpha[/itex] for integers n and m.
 
  • #4
You have expressions for f(x) [given], and f(x+p) [derived above].
Now assume that alpha is rational, m/n, and then see if there is any period p which satisfies the condition f(x)=f(x+p).
 

1. What does it mean for a function to be periodic?

A function is periodic if it repeats its values at regular intervals. This means that for every input value, the function will produce the same output value after a certain period of time or distance.

2. How do you prove that a function is periodic?

To prove that a function is periodic, you need to show that the function repeats its values at regular intervals. This can be done by looking for a pattern in the function's values or by using mathematical techniques such as finding the period or period shift of the function.

3. Can a non-periodic function be transformed into a periodic function?

Yes, a non-periodic function can be transformed into a periodic function by adding a periodic function to it. This process is known as periodic extension. It involves extending the domain of the function to make it periodic.

4. What are some common examples of periodic functions?

Some common examples of periodic functions include trigonometric functions such as sine, cosine, and tangent, as well as functions that represent physical phenomena, such as the position of a pendulum or the temperature over a day.

5. Can a function be both periodic and non-periodic?

No, a function cannot be both periodic and non-periodic as these are two mutually exclusive properties. A function can only be either periodic or non-periodic, but not both at the same time.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
488
  • Calculus and Beyond Homework Help
Replies
1
Views
722
  • Calculus and Beyond Homework Help
Replies
1
Views
253
  • Calculus and Beyond Homework Help
Replies
18
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
787
  • Calculus and Beyond Homework Help
Replies
3
Views
166
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
276
  • Calculus and Beyond Homework Help
Replies
2
Views
796
  • Calculus and Beyond Homework Help
Replies
14
Views
932
Back
Top