Find the period!

Algebra Level 5

f ( x , y ) = f ( ( 2 x + 2 y ) , ( 2 y − 2 x ) ) \large f(x,y) = f((2x+2y), (2y-2x))

For all real values of x x and y y , consider a periodic function f ( x , y ) f(x,y) that satisfies the equation above.

Suppose we define a function g g by g ( x ) = f ( 2 x , 0 ) g(x) = f(2^x,0) , what is the period of g ( x ) g(x) ?


The answer is 12.000.

Nishant Rai by Nishant Rai , 25, India

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1 solution

Otto Bretscher
May 7, 2016

Define a function h : C → R h:\mathbb{C}\rightarrow \mathbb{R} by h ( x + i y ) = f ( x , y ) h(x+iy)=f(x,y) . The given condition means that h ( z ( 2 − 2 i ) ) h\left(z(2-2i)\right) = h ( z ) =h(z) for all complex numbers z z . Applying this identity eight times, we see that h ( z ) = h ( z ( 2 − 2 i ) 8 ) h(z)=h\left(z(2-2i)^8\right) = h ( 2 12 z ) =h(2^{12}z) . In particular, g ( x ) = h ( 2 x ) = h ( 2 x + 12 ) = g ( x + 12 ) g(x)=h(2^x)=h(2^{x+12})=g(x+12) , so that g ( x ) g(x) has a period of 12 \boxed{12} .

3 Helpful 0 Interesting 0 Brilliant 0 Confused

I like your use of complex numbers. I just solved it using the matrix approach. By the way, my friend is buying me a copy of your Linear Algebra book as an early Christmas present. I'm excited to read it.

James Wilson - 3 years, 7 months ago

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