Laplace Transformation

Laplace Transform

Let f(t) be the function defined for all positive values of t , then Laplace Transform of f(t) is denoted as :

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

Some Examples :

(i) $\displaystyle L(1) = \frac{1}{s} $

(ii) $\displaystyle L(e^{a t}) = \frac{1}{s – a} $

(iii) $\displaystyle L(sin at) = \frac{a}{s^2 + a^2} $

(iv) $\displaystyle L(cos at) = \frac{s}{s^2 + a^2} $

Example : Find the Laplace Transform of cos2 t .

Solution: Since 1 + cos 2t = 2 cos2 t

$\displaystyle cos^2 t = \frac{1}{2}(1 + cos 2t) $

$\displaystyle L(cos^2 t) = L[\frac{1}{2}(1 + cos 2t)] $

$\displaystyle = \frac{1}{2}[L(1) + L(cos 2t)] $

$\displaystyle = \frac{1}{2}[\frac{1}{s} + \frac{s}{s^2 + 2^2}] $

$\displaystyle = \frac{1}{2}[\frac{1}{s} + \frac{s}{s^2 + 4}] $