LAPLACE TRANSFORMS

Defintion

: the Laplace transform of a function f(t) is

$${\cal L}\{f(t)\} = \int_0^\infty e^{-st}f(t)dt = F(s).$$

• The definition assumes that the integral exists.
• If
  a) f(t) is piecewise continuous for $t \geq 0$, and
  b) $\vert f(t)\vert \leq Ke^{at}$, for $t \geq M\geq 0$ and $K\geq 0$,
  then ${\cal L}\{ f \}$ exists for all s>a.

Properties of the Laplace Transform

• Linearity:
  
  $${\cal L}\{c_1f_1(t)+c_2f_2(t)\}=c_1{\cal L}\{f_1(t)\}+c_2{\cal L}\{f_2(t)\}.$$
  
  

• Transforms of Derivatives:
  
  $$\begin{eqnarray*}{\cal L}\{f'(t)\} & = &s{\cal L}\{f(t)\} - f(0),\\ {\cal L}\{f... ...f(t)\}-s^{n-1}f(0)-\\ & &\ \ \cdots-sf^{(n-2)}(0)-f^{(n-1)}(0). \end{eqnarray*}$$