Question

..
Determine the Fourier expansion of f(x) = x in
(-pie, pie)​

Answer 1

Answer:

Calculate Fourier series of f(x)=x2 , x∈ [−π,π] and determine module and phase spectrum

f(x)=a02+∑n=1+∞an cos(nx) + bn sin(nx)

a0=1π∫π−πx2 dx=23π2

an=1π∫π−πx2 cos(nx) dx=1π 2(π2n2−2)sin(nπ)+4πncos(nπ)n3=4(−1)nn2

bn=0∀n≥1

because f is even

f(x)=π23+4 ∑n=1+∞(−1)nn2 cos(nx)

How can I determine module and phase spectrum?

Should I calculate the Fourier series coefficients in different values of n, then calculate module and phase of the result?

Thanks!