MATH528

MATH528 Lesson15: Laplace transform of derivatives, integrals, ODEs

Definition. The Laplace transform of $f : [0, \infty) \to ℝ$ is

$F (s) = ℒ (f) (s) = \int_{0}^{\infty} f (t) e^{- s t} d t, F = ℒ (f),$

and $f (t) = ℒ^{- 1} (F) (t)$ , $f = ℒ^{- 1} (F)$ is the inverse Laplace transform.

Theorem. $ℒ (f^{'}) = s ℒ (f) - f (0)$ , $ℒ (f^{''}) = s^{2} ℒ (f) - s f (0) - f^{'} (0)$ ,

$ℒ (f^{(n)}) = s^{n} ℒ (f) - \sum_{j = 0}^{n - 1} s^{j} f^{(n - 1 - j)} (0)$

Theorem.

$ℒ (\int_{0}^{t} f (τ) d τ) = \frac{1}{s} F (s), \int_{0}^{t} f (τ) d τ = ℒ^{- 1} (\frac{1}{s} F (s))$

Example.

ℒ^{- 1} (\frac{1}{s^{2} + ω^{2}}) = \frac{\sin ω t}{ω} \Rightarrow ℒ^{- 1} (\frac{1}{s (s^{2} + ω^{2})}) = \int_{0}^{t} \frac{\sin ω τ}{ω} d τ = \frac{1 - \cos ω t}{ω^{2}}

In[1]:= InverseLaplaceTransform[1/s/(s^2+omega^2),s,t]

$\frac{1 - \cos (ω t)}{ω^{2}}$

In[2]:=