Periodic function of 2 variables

• Jul 22nd 2010, 10:03 AM
sashikanth
Periodic function of 2 variables
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.
• Jul 24th 2010, 03:41 AM
tonio
Quote:

Originally Posted by sashikanth
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.

$\displaystyle g(x+2)=f(2^{x+2},0)=f(2^{x+3},-2^{x+3})=f(2^{x+1},-2^{x+1})=f(2^x,0)=g(x)$

Tonio
• Jul 24th 2010, 03:54 AM
sashikanth
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. :)