Let $\displaystyle f(x,y) $ be a periodic function such that $\displaystyle f(x,y) = f((2x-2y),(2y-2x)) \forall \ x, \ y \epsilon R $. Define $\displaystyle g(x) = f(2^x,0) $. Prove that $\displaystyle g(x) $ is periodic and find its period.
Let $\displaystyle f(x,y) $ be a periodic function such that $\displaystyle f(x,y) = f((2x-2y),(2y-2x)) \forall \ x, \ y \epsilon R $. Define $\displaystyle g(x) = f(2^x,0) $. Prove that $\displaystyle g(x) $ is periodic and find its period.
Hi Tonio. Thank you for your reply. I am so sorry, I made an error in posting the question. The actual question in the book reads like this:
$\displaystyle f(x,y) = f((2x+2y),(2y-2x)) \forall \ x, \ y \ \epsilon \ R $
I do not understand what periodicity of a function of 2 variables means. I tried looking for the definition but I couldn't find it. Also, in your solution, I did not understand this step:
$\displaystyle f(2^{x+2},0)=f(2^{x+3},-2^{x+3}) $
I will be very grateful if you could explain it to me. Thank you.