Periodic function | Change of variables

[kingwinner]

"Is f(x)=sin(x²) periodic?
Answer: no."

WHY? I believe "sin" is always periodic? Can someone please explain?


Any help is appreciated!

 

[berkeman]

1) "Is f(x)=sin(x²) periodic?
Answer: no."

WHY? I believe "sin" is always periodic? Can someone please explain?



2) "Let f(x)=x.
Define the change of variable y=5x.
Then this implies g(y)=y/5.
[we have g(y)=f(x(y))=f(y/5) and f(x)=g(y(x))=g(5x)] "

If we define y=5x, then why does it imply g(y)=y/5? Shouldn't it be f(y)=y/5? Why do we need to introduce a new function g? (here we are doing a change of variable on the independent variable x, how come the dependent variable also changes?)
Also, WHY do we have g(y)=f(x(y)) and f(x)=g(y(x))? I don't understand this.


Any help is appreciated!

On the first one, what is the definition of periodic? And when you have a non-linear argument to the sin() funtion, what is the period?

 

[HallsofIvy]

If x= 0, [itex]sin(x^2)= sin(0^2)= sin(0)= 0[/itex]. When is [itex]sin(x^2)= 0[/itex] again? Is that a period?

 

[kingwinner]

On the first one, what is the definition of periodic? And when you have a non-linear argument to the sin() funtion, what is the period?

I know that f(x)=sin(x) has period 2pi, g(x)=sin(2x) has period pi, etc. Since I am seeing sin in the function sin(x^2), this leads me to think that sin(x^2) is periodic as well.

For sin(x^2), The zero set is {x: x^2 = (n)(pi)}, but how can I know whether it's periodic or not?
I think it's hard for me to tell whether a function given randomly to me is periodic or not. Is there any systematic way to answer this? I am not sure where to start...

Thanks!

 

[kingwinner]

If x= 0, [itex]sin(x^2)= sin(0^2)= sin(0)= 0[/itex]. When is [itex]sin(x^2)= 0[/itex] again? Is that a period?

For sin(x^2), The zero set is {x: x^2 = (n)(pi)}, but how can I know whether it's periodic or not? I just can't tell...

 

[lanedance]

periodic in my book would be
f(t+a) = f(t) for all t, for som constant a

in your case, solve for the first few zeros, and see what the difference bewteen them is (cf with a in the above) , see if there's any pattern which you can pick out which shows its not periodic