Prove that $\cos\sqrt x$ is not a periodic function.
My solution goes like this:
If $\cos\sqrt x$ is a periodic function , then let $T$ be its period . Now, $\cos\sqrt x=\cos\sqrt {x+T}$ . So, we get, $\sqrt {x+T}\pm\sqrt {x}=2k\pi$. Now, for a particular $k\in\mathbb {Z}$ , this identity is impossible because the left member is a variable continous argument in $x$ , while its right member is a constant. Hence, $\cos\sqrt x$ is not a periodic function.
Is the above solution correct? Is it valid? If not, where is it going wrong?…
Best Answer
The statement is equivalent to:
This also implies that, $T>0$.
We claim that, such constant $T≠0$ doesn't exist.
Proof: Let $x=0$. Then you have:
$$\begin{align}&\cos \sqrt T=1\\ \implies &\sqrt T=2πn,\; n\in\mathbb Z^{+}\\ \implies &T=4π^2n^2,\;n\in\mathbb Z^{+}.\end{align}$$
Now, let $x=4π^2$. We obtain:
$$ \begin{align}&\cos 2π=\cos \sqrt {4π^2+T}=1\\ \implies &\cos 2π\sqrt {n^2+1}=1\\ \implies &\sqrt {n^2+1}=k,\;k\in\mathbb Z^{+}\\ \implies &n^2+1=k^2\\ \implies &(k-n)(k+n)=1\\ \implies &n=0 \;\text{or}\; T=0\\ &\text {A contradiction .}\end{align} $$
Explanation:
Just because you missed a small detail, your proof couldn't work.
You derived the following relationship:
$$\sqrt {x+T}\pm\sqrt {x}=2kπ,\,k\in\mathbb Z$$
Then, that's correct and you are right.
But observe that, this doesn't imply us, you can consider $2πk$ as a constant for a particular $k\in\mathbb Z$.
This implies that,
In other words, you can also understand this statement as follows:
If $\cos \sqrt x=\cos \sqrt{x+T}$, then
$$\sqrt {x+T}\pm\sqrt {x}=2\pi k,\,k\in\mathbb Z$$
holds, for some $k\in\mathbb Z$.
Therefore, we cannot say that the right-hand side should be a constant even for a particular $k\in\mathbb Z$.
For instance, you can take
$$\cos 2π=\cos 4π=1$$
This implies,
$$4π-2π=2π×\color {red}{1}$$
or
$$4π+2π=2π×\color {red}{3}$$
Now take,
$$\cos 2π=\cos 6π=1$$
This yields,
$$6π-2π=2π×\color {red}{2}$$
or
$$6π+2π=2π×\color {red}{4}$$
We see that, since $k$ is not a constant, we cannot consider the right-hand side as a constant for any particular $k\in\mathbb Z$.