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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 $\Bbb R$.

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

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