UNIT STEP FUNCTION

Solution: Example :

Ex: Write the following function in terms of the unit step function

Ex1: Ex2: Ex3: = ?

Ex: Show that L{ f(t)u(t-a) }=e -as L{ f(t+a) } Proof: L(f(t-a)u(t-a)}=e -as F(s) (1) Let f(t-a)=g(t) then f(t)=g(t+a), put in (1) L{g(t)u(t-a)}=e -as L{g(t+a)}, change from g to f simply L{f(t)u(t-a)}=e -as L{f(t+a)}, Ex:

Impulse Function Define the function f k (t-a) as In terms of unit step functions Dirac delta function or unit impulse function

Mathematical expression for the unit impulse function Some properties of the unit impulse function a)b) c)

Ex: Solve Solution: Taking the laplace transform of both sides y(0)=0, y’(0)=0

Convolution Convolution of f(t) and g(t), h(t) is defined as: Convolution Theorem Let H(s), F(s), and G(s) denote the laplace transforms of h(t), f(t), and g(t). If h is the convolution of f and g, h=f * g then H(s)=F(s)G(s) h(t)=L -1 {F(s)G(s)}

Example:

Laplace transform of tf(t)

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