The Fourier series represents a periodic function as a descrete vectors. The Fourier transformation turns a time domain non-periodic function into a frequency domain continuous function. The Fourier series and transformation change a single time base \(t\) into infinite frequency basis \(e^{inx}\) or \(e^{iwx}\). The function on infinite basis domain can be represented by a vector or a function of basis domain \(v_{n}\) or \(f(w)\). This is a coefficients of Fourier series or Fourier transformation.

The basis of Fourier transformation is pure frequency \(e^{iw}\). The domain of Laplace transfomation is frequency \(w\) and damping component \(\sigma\) which compose damping ocilation function, \(e^{s} = e^{(iw+\sigma)}\). The function which represent Laplace transformation \(F(s)\) is a function of complex domain \(s\). The Fourier transformation is a special Laplace transformation of no damping term \(s = 0 \cdot \sigma +iw\).

The **periodic** function can be represented by a series not a continuous function. A condition makes a function can be represented by pure frequency domain i.e. Fourier transformation, not a complex domain i.e. Laplace transformation. The condition is

from wikipedia https://en.wikipedia.org/wiki/Laplace_transform#Fourier_transform

$$\[\begin{align} \widehat{f}(\omega) &= \mathcal{F}\{f(t)\} \\[4pt] &= \mathcal{L}\{f(t)\}|_{s = i\omega} = F(s)|_{s = i \omega} \\[4pt] &= \int_{-\infty}^\infty e^{-i \omega t} f(t)\,dt~. \end{align}\]

Laplace transformation makes a differential equation to an algebra equation.

\[Laplace transformation\]

\[ \mathcal{L}[f(t)] = F(s) = \int_{t=0}^{\infty} f(t)e^{-st}dt \]

\[Transfer function\]

\[ H(s) = Y(s)/X(s) \] \[ Y(s) = H(s)X(s) \]

where \(Y(s)\) and \(X(s)\) are Laplace transformed \(y(t)\), i.e. solution and \(f(t)\) i.e. input.

The \(Y(s)\) is a function of \(s\) which represents coefficients of damped frquency basis \(e^{\sigma + iw}\). We are not looking for the solution \(s\) for the \(Y(s)\). We are looking for the inverse Laplace transformation of \(Y(s)\). The inverse Laplace transformation turns a function \(Y(s)\) with infinite damped frquency basis \(e^{\sigma + iw}\) to the solution of linear differential equation \(y(t)\) that is a function with a single domain basis \(t\).

The Laplace transformation has poles that blow up at a point. The poles were determined by constants of differential equation and the input term.