Question

You have given a function λ : R → R with the following properties (x ∈ R, n ∈ N):

λ(n) = 0 , λ(x + 1) = λ(x) , λ(n+1/2)=1
Find two functions p, q : R → R with q(x)6=0 for all x such that λ(x) = q(x)(p(x) + 1).

Answer 1

The required function is,

$\lambda(x) = \dfrac{1-cos(2πx)}{2}$

Comparing the function to $\lambda(x) = q(x)(p(x) + 1)$

$\therefore p(x) = -cos(2πx),\\ and\ q(x) =\frac{1}{2}$

For x = 1,

$\lambda(1) = \dfrac{1-cos(2π(1))}{2}$

$\lambda(1) = \dfrac{1-cos(2π)}{2} = 0$

For x= 2,

$\lambda(2) = \dfrac{1-cos(2π(2))}{2}$

$\lambda(2) = \dfrac{1-cos(4π)}{2} = 0$

$\therefore$ when x = n,

λ(x) = λ(n) = 0

and λ(n) derived is;

$\lambda(n) = \dfrac{1-cos(2π(n))}{2}$

$\lambda(n) = \dfrac{1-cos2nπ}{2}$