Half range Series and Harmonic Analysis

1. Find the Half range cosine series for f(x)=x in (0,π) . 2. Find the half range cosine series for f(x)=x(π-x) in (0,π) 3. Find the Half range cosine series of f(x)=(π-x)2, 0<x<π. Hence find the sum of series 4. Find the half range cosine series expansion of (x-1)2 in 0<x<1. 5. Obtain the Fourier cosine series expansion of f(x) =x in 0<x<4. Hence deduce the value of a. 6. Find the Fourier series of y=f(x) up to third harmonic which is defined by the following data in (0,2π) x 0 π/3 2π/3 π 4 π/3 5 π/3 2 π f(x) 1 1.4 1.9 1.7 1.5 1.2 1 7. Find the Fourier cosine series up to third harmonic to represent the function given by the following data: x 0 1 2 3 4 5 y 9 18 24 28 26 20 8. Determine the first two harmonics of Fourier series for the following data. x 0 T/6 T/3 T/2 2T/3 5T/6 T f(x) 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98