If $f(x)f(y)=f(\sqrt{x^2+y^2})$ how to find $f(x)$

If you set $$g(x) := f(\sqrt{x})$$ for $x \in [0, \infty)$ then you get $$g(x)g(y) = f(\sqrt{x})f(\sqrt{y}) = f(\sqrt{x+y}) = g(x+y)$$

You see that $g(x) \geq 0$, and if $g(x) = 0$ for some $x > 0$ then $g \equiv 0$. Thus, you can look at $$h(x) := \log(g(x))$$ It satisfies $$h(x) + h(y) = \log(g(x)g(y)) = \log(g(x+y)) = h(x+y)$$ If you impose any reqularity condition on $f$ you can think of, you will get $h(x) = \alpha x$, and consequently $$f(x) = \exp(\alpha x^2)$$ for $x > 0$. You can generalise this result to $x < 0$ using the fact that from the initial equation it follows that $f$ is even.

Change variable, $g(u) = f(\sqrt{u})$. You need to decide what you want for negative $u$. Then this functional equation becomes $g(u+v)=g(u)g(v)$.

Let' assume you want solution to $f:\mathbb{R}\mapsto\mathbb{R}$ that satisfies this equation $$\forall x,y\in\mathbb{R},f(x)f(y)=f\left(\sqrt{x^2+y^2}\right)\tag{1}$$

Let's plug in $(x,y)=(0,0)$ in the equation $(1)$, $$f(0)^2=f(0)\Rightarrow f(0)\in\{0,1\}$$

Case-1 : $f(0)=0$

For $x>0, f(x)=f\left(\sqrt{x^2+0^2}\right)=f(x)f(0)=0$.

For $x<0, f(x)^2=f(x)f(x)=f\left(\sqrt{x^2+x^2}\right)=f\left(\sqrt{2}|x|\right)=0\Rightarrow f(x)=0$.

So, if $f(0)=0$, then $\forall x\in\mathbb{R}, f(x)=0$

Case-2 : $f(0)=1$

In this case, we claim that this function must be even.

Proof :

$f(x)=f(x)f(0)=f\left(\sqrt{x^2+0^2}\right)=f\left(\sqrt{(-x)^2+0^2}\right)=f(-x)f(0)=f(-x)\tag{2}$

Now, we will consider two assumptions under this case.

Assumption-A :

$\exists a>0, f(a)=0$.

Then, $\forall x\geq a$, $$f(x)=f\left(\sqrt{a^2+\left(\sqrt{x^2-a^2}\right)^2}\right)=f(a)f\left(\sqrt{x^2-a^2}\right)=0$$

Assumption-B :

$\exists b>0, f(b)>0$

Then, $$f\left(\sqrt2b\right)=f\left(\sqrt{b^2+b^2}\right)=f(b)^2>0$$ Proceeding this way, we can find arbitrarily large real number $x$ for which $f(x)>0$. Which contradicts with Assumption-A that says after reaching a bound, we can't find and real number $x$ for which $f(x)>0$.

So, we conclude that $a$ and $b$ can't exist simultaneously. If we assume the existence of $a$, We get the solution : $$f(x) =\left\{\begin{array}{10}1 & \mbox{if } x=0\\0 & \mbox{otherwise}\end{array}\right.$$ We can check and indeed, this is a valid solution.

Now we assume the existence of $b$, that is, $\exists b>0, f(b)>0$. Since this assumption excludes the possibility of the existence of real number $a$ for which $f(a)=0$, we only have two options : either $f(x)>0$ or $f(x)<0$ for any real number $x$. We claim that $\forall x, f(x)>0$.

Proof : $$f(x)=f\left(|x|\right)=f\left(\sqrt{\left(\frac{x}{\sqrt2}\right)^2+\left(\frac{x}{\sqrt2}\right)^2}\right)=\left(f\left(\frac{x}{\sqrt2}\right)\right)^2>0$$ So, $\forall x, f(x)>0$.

Now, we will plug in $(\sqrt{x},\sqrt{y})$ where $x,y\geq0$ in equation $(1)$, $$f\left(\sqrt{x}\right)f\left(\sqrt{y}\right)=f\left(\sqrt{x+y}\right)\\\Leftrightarrow \ln\left(f\left(\sqrt{x}\right)\right)+\ln\left(f\left(\sqrt{y}\right)\right)=\ln\left(f\left(\sqrt{x+y}\right)\right)$$

Let's define $\phi:\mathbb{R}_{\geq0}\mapsto\mathbb{R}$ such that $\phi(x)=\ln\left(f\left(\sqrt{x}\right)\right)$. Note that $\phi(x)+\phi(y)=\phi(x+y)$ which satisfies the Cauchy's functional equation. So, $\phi$ is a solution to Cauchy's functional equation. The most trivial solution is $\forall x\geq0,\phi(x)=kx$ where $k\in\mathbb{R}$. There are non trivial solutions which are highly pathological functions.

So, analyzing all the cases we got this solutions :

1. $\forall x, f(x)=0$
2. $f(x) =\left\{\begin{array}{10}1 & \mbox{if } x=0\\0 & \mbox{otherwise}\end{array}\right.$
3. $\forall x, f(x)=e^{\phi(x^2)}$ where $\phi:\mathbb{R}_{\geq0}\mapsto\mathbb{R}$ and $\phi$ satisfies Cauchy's functional equation.