Find It's Period.
is a non-zero function defined from the integers to the reals, satisfying
Find the fundamental period of .
The answer is 12.
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1 solution
Let us express f(x+1) + f(x-1) = sqrt(3)*f(x) as the characteristic equation:
r^2 - sqrt(3)*r + 1 = 0 (i)
which yields the complex conjugate pair of roots: r = sqrt(3)/2 (+/-) i/2, or r = exp(i * pi/6), exp(-i * pi/6). Putting these roots together now gives an expression for f(x):
f(x) = A [exp(i * pi/6)]^x + B [exp(-i * pi/6)]^x,
and if one chooses A = B = 1/2, then one can obtain the Eulerian Cosine:
f(x) = (1/2) exp(i * pi x/6) + (1/2) exp(-i * pi x/6) = cos(pi*x / 6).
Hence the fundamental period equals 2*pi / (pi/6) = 12.