# functional equations

Find all solutions $f:\Bbb R^2 \to \Bbb R$ satisfying $$f(xu-yv, yu+xv)=f(x, y)f(u, v).$$

Solution of the following equation $$f(xu+yv, yu-xv)=f(x, y)f(u, v)$$ is known as $$f(x,y)=m(x^2+y^2),$$ where $m$ is multiplicative function on $\Bbb R$.

Hint: As a complex function $f : \mathbb{C} \to \mathbb{R}$, you have $f(zz')=f(z)f(z')$.