The Formulas for Fourier Series
$ x(t) = \sum^{\infty}_{k = -\infty} a_k e^{jk\pi t}\, $
where $ a_k=\frac{1}{T} \int_0^Tx(t)e^{-jk\omega_ot}dt $
Chosen Formula
$ x(t) = (5+3j)cos(4t) + (1+2j)sin(3t) $
Computation
First we want to compute the period (T) for this function. The period of sin and cos is $ 2\pi $, therefore the combined period is also $ 2\pi $.
Next we compute the coefficients. Since we are using sin and cos, we can use their equivalent formulas.
$ x(t) = (5+3j)cos(4t) + (1+2j)sin(3t) $
$ = (5+3j)(\frac{e^{4jt} +e^{-4jt}}{2}) + (1+2j)(\frac{e^{j3t} - e^{-j3t}}{2j}) $
$ = \frac{5+3j}{2}e^{4jt} + \frac{5+3j}{2}e^{-4jt} + \frac{1+2j}{2j}e^{j3t} - \frac{1+2j}{2j}e^{-3jt} $
multiplying by complex conjugate we get:
$ = \frac{5+3j}{2}e^{4jt} + \frac{5+3j}{2}e^{-4jt} + (1-\frac{j}{2})e^{j3t} - (1-\frac{j}{2})e^{-j3t} $
since $ \omega_0=\frac{2\pi}{T} $ and $ T=2\pi $ then $ \omega_0=1 $
Therefore for the first term k=4 and for the second term k=-4. Likewise, for the third and fourth terms k=3 and k=-3 respectively. Therefore our coefficients are:
$ a_3 = a_{-3} = \frac{5+3j}{2} $
And $ a_3 = a_{-4} = (1-\frac{1}{2}) $
And $ a_k = 0 $ else
