Inverse Laplace to Fourier series

[Debdut]

I have the following laplace function
F(s) = (A/(s + C)) * (1/s - exp(-sα)/s)/(1 - exp(-sT))

I think that the inverse laplace will be-
f(t) = ((A/C)*u(t) - (A/C)*exp(-Ct)*u(t)) - ((A/C)*u(t-α) - (A/C)*exp(-C(t-α))*u(t-α))
and
f(t+T)=f(t)

Now I want to find the Fourier series expansion of f(t) and find the magnitudes of sin(2πt/T) and cos(2πt/T), how should a₀, a_n, b_n be defined, I mean what will the integration limits?

 

[jvicens]

Hello,

Thank you for your post. It looks like you have correctly found the inverse Laplace transform of the given function. To find the Fourier series expansion of f(t), we can use the following equations:

a0 = (1/T) * ∫f(t)dt from 0 to T
an = (2/T) * ∫f(t)*cos(n*2πt/T)dt from 0 to T
bn = (2/T) * ∫f(t)*sin(n*2πt/T)dt from 0 to T

In this case, the integration limits will be from 0 to T, as we are looking at the periodicity of the function over one period. The magnitudes of sin(2πt/T) and cos(2πt/T) can be found by taking the square root of the sum of the squares of the coefficients an and bn.

I hope this helps. Let me know if you have any further questions.