Question

Why is sin(x^2) not a periodic function?

Answer 1

While `sin(x)` repeats every time `x` changes by a fixed amount (`2pi`), `sin(x^2)` repeats only when `x^2` changes by `2pi`, and this does not happen at uniform intervals of `x`.

Explanation:

For a function `f(x)` to be periodic with a period `X`, we must have

`f(x+X)=f(x) qquad forall x`

This immediately implies that `f(x), f(x+X), f(x+2x), f(x+3X),...` etc. all have the same value. So, a periodic function of `x` repeats after equally spaced values of `x`.

Now, `sin(x^2)` attains the value 0, for example, when `x^2` attains the values 0, `pi`, `2pi`, `3pi`, etc. While these values are equally spaced, the corresponding `x` values are 0, `sqrt(pi)`, `sqrt(2pi)`, `sqrt(3pi)` etc. - and these are not equally spaced.

Answer 2

While `sin(x)` repeats every time `x` changes by a fixed amount (`2pi`), `sin(x^2)` repeats only when `x^2` changes by `2pi`, and this does not happen at uniform intervals of `x`.

Explanation:

For a function `f(x)` to be periodic with a period `X`, we must have

`f(x+X)=f(x) qquad forall x`

This immediately implies that `f(x), f(x+X), f(x+2x), f(x+3X),...` etc. all have the same value. So, a periodic function of `x` repeats after equally spaced values of `x`.

Now, `sin(x^2)` attains the value 0, for example, when `x^2` attains the values 0, `pi`, `2pi`, `3pi`, etc. While these values are equally spaced, the corresponding `x` values are 0, `sqrt(pi)`, `sqrt(2pi)`, `sqrt(3pi)` etc. - and these are not equally spaced.