Laplace Transforms

Question 1

Definition

Answer 1

F(s) = ∫[∞,0] e^(-st)*f(t) dt

Where s is a complex variable.

Question 2

Linearity Property

Answer 2

L{αf(t) + βg(t)} = αL{f(t)} + βL{g(t)}

Question 3

Differentiation Property

Answer 3

L{df/dt} = sF(s) - f(0)
L{d²f/dt²} = s²F(s) - sf(0) - f’(0)

Question 4

Sifting Property

Answer 4

L{δ(t-T)} = e^(-st)
L{H(t-T)} = e^(-st)/s

Question 5

Frequency Shift Property

Answer 5

L{e^(-kt)*f(t)} = F(s + k)

Question 6

Time Delay Property

Answer 6

L{f(t-a)H(t-a)} = e^(-as)F(s)

Question 7

Scaling Property

Answer 7

L{f(at)} = (1/a)*F(s/a)

for a > 0

Question 8

What does convolution integral do?

Answer 8

It passes one function through another and measures there degree of ‘interaction’

Question 9

Convolution Theorem

Answer 9

L{f(t)*g(t)} = L{f(t)} * L{g(t)}
= F(s) * G(s)

If f(t) & g(t) are causal functions (f(t) = g(t) = 0)

Question 10

Steps to use Laplace to solve an ODE

Answer 10

1) Take LT of both sides
2) Rearrange to make F(s) the subject
3) Use known LT’s to do inverse and find f(t)