# 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 $$f(w)$$. 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

\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.

##### Jun Kang
###### Clinical Assistant Professor of Hospital Pathology

My research interests include pathology, oncology and statistics.