I'm new to Fourier series, and in a section of Laplace transform I found this problem:
Determine the Fourier's trigonometric series of odd extension of the function $$f(t)=\left\lbrace -1\ \ if\ \ -\pi<t<0 \atop 1\ \ if \ \ 0<t<\pi \right.$$
Could anyone provide me some hints on what to calculate? and its relation to Laplace if any?
Solution After following the sugestion I found the numeric coefficients $a_0=0$ $a_k=0$, $b_k=\frac{2-2\cos(k\pi)}{k\pi}=\frac{2(1-(-1)^k)}{k\pi}$ and then $$f(t)=\sum_{k=1}^\infty \frac{2}{\pi k}(1-(-1)^k)\sin(kt)$$
Best Answer
This answer assumes $f(t)$ is periodic as defined in formula (1) below.
$$f(t)=\left\{\begin{array}{cc} 1 & 0\leq t<\pi \\ -1 & \pi \leq t<2 \pi \\ f(t \bmod (2 \pi )) & \text{otherwise} \\ \end{array}\right.\tag{1}$$
Since $f(t)$ is an odd function of $t$ it can be represented by a Fourier sine serie as illustrated in formula (2) below.
$$f(t)=\underset{N\to\infty}{\text{lim}}\left(-\frac{2}{\pi}\sum\limits_{n=1}^N\frac{(-1)^n-1}{n}\,\sin(n\,t)\right)\tag{2}$$
Figure (1) below illustrates formula (2) for $f(t)$ evaluated at $N=100$ in orange overlaid on the blue reference function defined in formula (1). The red discrete evaluation points illustrate the evaluation of formula (2) at multiples of $\pi$.
Figure (1): Illustration of formula (2) for $f(t)$ in orange overlaid on blue reference function defined in formula (1)
The Laplace transform of $f(t)$ is illustrated in formula (3) below, and Mathematica indicates formula (3) below simplifies to formula (4) below.
$$\mathcal{L}_t[f(t)](y)=\int\limits_0^\infty f(t)\,e^{-t\,y}\,dt=\underset{N\to\infty}{\text{lim}}\left(-\frac{2}{\pi}\sum\limits_{n=1}^N\frac{(-1)^n-1}{n^2+y^2}\right)\tag{3}$$
$$\mathcal{L}_t[f(t)](y)=\frac{\tanh\left(\frac{\pi\,y}{2}\right)}{y}\tag{4}$$
Figure (2) below illustrates formula (3) for $\mathcal{L}_t[f(t)](y)$ evaluated at $N=100$ in orange overlaid on the blue reference function defined in formula (4).
Figure (2): Illustration of formula (3) for $\mathcal{L}_t[f(t)](y)$ in orange overlaid on blue reference function defined in formula (4)