Laplace Transform¶

Definition¶

The Laplace Transforms of Basic Functions¶

Constraints¶

• constraint 1 For a function $f(t)$, $\mathrm{\exists }$ constant $c$, $M>0$, and $T>0$ s.t.

  $$|f(t)|\le M{e}^{ct},\mathrm{\forall }t>T$$

  $\to f(t)$ 不能跑得比 $\exp (ct)$ 快

• constraint 2

  $f(t)$ should be piecewise continuous on $[0,\mathrm{\infty })$

  $\to \mathrm{\forall }t\in [0,\mathrm{\infty })$, $f(t)$ 為 discontinuous 的點有限

note : these constraints are sufficient conditions

五大定理

Five Theorem¶

Transforms of Derivatives¶

First Translation Theorem¶

translation for $s$

proof :

Second Translation Theorem¶

translation for $t$

or

in which $u(t)$ is an unit step function.

proof :

Derivatives of Transforms¶

Convolution¶

Definition :

When $f(t)=0$ for $t<0$ and $g(t)=0$ for $t<0$

We have the same definition as below.

Convolution Theorem¶

Integration¶

from theorem above

Transform of a Periodic Function¶

When

then

來自無窮等比級數

Dirac Delta Function¶

Integration¶

Sifting¶

Laplace Transform¶

proof :

due to second translation theorem, we have