2
$\begingroup$

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 $\Bbb R$.

$\endgroup$
1
  • $\begingroup$ The constant function $ f ( x , y ) = 1 $ also satisfies your second functional equation. $\endgroup$ Nov 10, 2020 at 23:27

1 Answer 1

2
$\begingroup$

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

$\endgroup$
0

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .