# Fourier series representation help

#### fdsa

Joined Aug 16, 2011
9
I'm trying to find the fourier series representation of the following periodic function: $$x(t)=cos(t)+sin(2t)+cos(3t-\frac{\pi}{3})$$

So I've found:
$$c_1 = c_{-1} = 1/2$$
$$c_2 = -\frac{j}{2}$$
$$c_{-2} = \frac{j}{2}$$

$$c_3 = \frac{1}{2}\times e^{-j\pi / 3}$$
$$c_{-3} =\frac{1}{2}\times e^{j\pi / 3}$$

According to the solutions the answer is supposed to be:
$$c_3 = c_{-3}= 1/2$$

Did I do something wrong?

#### anhnha

Joined Apr 19, 2012
884
Here the formula of Fourier series:

$$x(t) = \sum_{- \propto }^ \propto c_{n} e^{jn \omega _{0} t}$$

And by using Euler formula:

$$cos( \varphi ) = \frac{e^{j \varphi } + e^{-j \varphi }}{2}$$

$$sin( \varphi ) = \frac{e^{j \varphi } - e^{-j \varphi }}{2j}$$

Now we can rewrite x(t) as follows:

$$x(t) = \frac{e^{jt} + e^{-jt}}{2} + \frac{e^{j2t} - e^{-j2t}}{2j} + \frac{e^{j(3t - \frac{ \Pi }{3}) } + e^{-j(3t - \frac{ \Pi }{3} )}}{2}$$

And according to that result, you are correct.