How to find out if a function is periodic or not?

[gafar]

I need to find if Sin(x^2) is a periodic function. As I think its not periodic but I need to proof that.
I know that its possible to use f(x) = f(x+T), while T is the period frequency.
But how to find out T ? and how to contradict this equation to say that the function is not periodic.

Thanks, gafar.

 

[Rajini]

Hi gafar,
i quickly plot your function f(x)=sin(x*x).
between three x ranges [see figures below]
1. between -10 to 10,
2. -100 to 100 and
3. -1000 to 1000.
from that it follows a kind of periodic change. [in general most of the sin functions are periodic]

 

[mikeph]

I guess one sloppy proof would be that this is kinda like a sin(w*x) graph where w is the frequency, and w = x, ie. the frequency increases with x. So nowhere can this be periodic as the frequency of the wave is different at each point.

 

[gafar]

Hi gafar,
i quickly plot your function f(x)=sin(x*x).
between three x ranges [see figures below]
1. between -10 to 10,
2. -100 to 100 and
3. -1000 to 1000.
from that it follows a kind of periodic change. [in general most of the sin functions are periodic]

Thanks, still waiting for your pics to be approved. So just to be sure, when you see periodic changes it means that the function is NOT periodic right?
And I know that its easy to conclude the answer from a plot but is there an simple arithmetic proof?
and thanks a lot.!

 

[mikeph]

How's this :

assume it is periodic...
for the function to be periodic with period T, y(x) = y(x+T)
but also as we know y is continuous, y'(T) = y'(x+T) (same gradient for it to be periodic)

ie.

sin(x^2) = sin(x^2 + 2xT + T^2)
2x*sin(x^2) = (2x+2T)sin(^2 + 2xT + T^2)

divide bottom by the top:

2x = (2x+2T)
T = 0

ie. it is not periodic!

 

[Rajini]

Hi,
When a function changes periodically, that means the function is periodic.
Your function to me is some kind of periodic motion, but not just noise!
There are many types of periodic motion..simplest one is sin(x) function..
If a function is not periodic= probably should be noise..
Sorry i am not good in mathematical proof..however, someone will help you..
good luck

 

[Rajini]

Mikey,
i notice some typing error..
will it is like this:
$$\sin (x^2)=\sin(x^2+2xT+T^2)$$
$$2x\cos(x^2)=2(x+T)\cos(x^2+T^2+2xT)$$??

 

[mikeph]

Oh god. I am not on the ball today. Sorry.

 

[gafar]

thanks guys but actually I am little confuse now because one says its periodic and other says its not.

 

[Rajini]

Gafar,
Please prove some details..where you want to use sin(x*x)?? etc..
As mikey said:
if T is same then it is periodic..otherwise not..Remember sine wave in which the period is same, i.e. the wavelength is same..so sine wave is periodic..
[why no reply from experts?]

 

[willem2]

thanks guys but actually I am little confuse now because one says its periodic and other says its not.

There's a definition of a periodic function. Who uses the definition and who has just
plotted it and said that it looks periodic?

The proof by Mikey is incomplete however.
It can be made to work if you substitute x = 0. You get

$\sin {(T^2)} = 0$
$2 T \cos {(T^2)} = 0$

Since sin(x) and cos(x) are never both 0, these equations can only be both fulfilled
if T = 0

 

[Rajini]

Hi Willem,
so sin(x*x) is periodic if T is not equal to zero!
is that correct?
I am just asking out of curiosity.
thanks

 

[mikeph]

What the proof (mine failed but willem2's works) shows is that if you begin by assuming the function is periodic, then the period must be 0. But a period of 0 does not make sense so the function cannot be periodic.

 

[Mentallic]

Hi gafar,
i quickly plot your function f(x)=sin(x*x).
between three x ranges [see figures below]
1. between -10 to 10,
2. -100 to 100 and
3. -1000 to 1000.

Rajini the function y=sin(x²) doesn't look at all like what you're seeing in those graphs! The fluctuations of 1 period at x=1000 are approx 6x10^-3 which is tiny! In other words, the computer (ignoring miscalculations) doesn't have enough pixels to represent the vast number of up-downs of the function so it has given you what you see there.

 

[Rajini]

Aha, i noticed that problem...
I made those plots using gnuplot.
If that function is periodic then the period should be extremely small!