How to derive the complex Fourier series of $s(t) = 1-e^{-2t}$?

Question

I have the periodic function $s(t)=1-e^{-2t}$. I am required to derive the complex Fourier series of $s(t)$.

I have some knowledge of Fourier series but not enough to know if I am doing it correctly. If there is other information needed just ask.

Answer 1

The coefficients of the complex Fourier Series are given by

$$c_n=\frac{1}{T}\int_0^Ts(t)e^{i2\pi n t/T}dt$$

For $s(t)=1-e^{-2t}$ with a period $T=5$, we have

$$\begin{align} c_n&=\frac{1}{5}\int_0^5 (1-e^{-2t})\,e^{i2\pi n t/5}dt\\\\ &=\delta_{n,0}+\frac12\,\frac{e^{10}-1}{e^{10}(-5+i\pi n)} \end{align}$$

where $\delta_{n,m}$ is the Kronecker Delta and equals $1$ for $n=m$ and equals $0$ when $n \ne m$.

Thus, we have

$$s(t)=\sum_{-\infty}^{\infty}\left(\delta_{n,0}+\frac12\,\frac{e^{10}-1}{e^{10}(-5+i\pi n)}\right)\,e^{i2\pi nt/5}$$