Laplace transformation

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 $f\left(w\right)$. 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 $F\left(s\right)$ 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

$$$$\begin{array}{cc}\hat{f}\left(\omega \right)& =F\left\{f\right(t\left)\right\}\\ & =L\left\{f\right(t\left)\right\}{|}_{s=i\omega }=F\left(s\right){|}_{s=i\omega }\\ & ={\int }_{-\infty }^{\infty }{e}^{-i\omega t}f\left(t\right)dt.\end{array}$$

Laplace transformation makes a differential equation to an algebra equation.

$$Laplacetransformation$$

$$L\left[f\right(t\left)\right]=F\left(s\right)={\int }_{t=0}^{\infty }f\left(t\right)e^{-st}dt$$

$$Transferfunction$$

$$H\left(s\right)=Y\left(s\right)/X\left(s\right)$$ $$Y\left(s\right)=H\left(s\right)X\left(s\right)$$

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

The $Y\left(s\right)$ 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 $Y\left(s\right)$. We are looking for the inverse Laplace transformation of $Y\left(s\right)$. The inverse Laplace transformation turns a function $Y\left(s\right)$ with infinite damped frquency basis $e^{\sigma +iw}$ to the solution of linear differential equation $y\left(t\right)$ 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.