Find the complex Fourier series

Question

Find the complex Fourier series representation of the function

$$ f(t) = \begin{cases} 1,\quad\text{if}\quad 0 < t < 2 \\ 0,\quad\text{if}\quad 2 < t < 4 \end{cases} $$ with the period 4.

Answer 1

The coefficients are

$$c_{n} = \frac{1}{4}\int_{0}^{2}e^{- i\frac{\pi n x}{2}}dx = \frac{i}{2 \pi n}(e^{- i\pi n}-1),$$ for $n \in \mathbb{Z}-\{0\}.$

For $n = 0:$

$$c_{0} = \frac{1}{4}\int_{0}^{2}dx \Rightarrow c_0 = \frac{1}{2}.$$

For $n$ odd:

$$c_{n} = \frac{i}{2 \pi n}(e^{- i\pi n}-1)= -\frac{i}{\pi n}.$$

For $n$ even, and $n >0$:

$$c_{n} = \frac{i}{2 \pi n}(e^{- i\pi n}-1)= 0.$$

Thus, $$f(x) =\sum_{n=-\infty}^{\infty}c_ne^{\frac{i\pi n x}{2}}.$$