Write the Fourier series representation for the periodic function f .One period of the
function f is defined as below.
To find the Fourier series representation for a periodic function, we need to determine the coefficients of the Fourier series, which are the amplitudes of the cosine and sine terms.
The Fourier series representation of a periodic function f(x) with period T is given by:
f(x) = a₀/2 + ∑ [aₙ cos(nω₀x) + bₙ sin(nω₀x)]
Where:
- a₀ is the DC component, equal to the average value of the function over one period.
- aₙ and bₙ are the Fourier coefficients. These can be calculated using the formulas:
aₙ = (2/T) ∫[f(x) cos(nω₀x) dx] (for n > 0)
bₙ = (2/T) ∫[f(x) sin(nω₀x) dx]
- ω₀ = 2π/T is the fundamental frequency of the function.
To find the Fourier series representation for the given function f, we also need to know the expression of f(x). Please provide the expression of one period of the function f, and I can guide you on how to calculate the coefficients a₁, b₁, a₂, b₂, etc.