UNIT STEP FUNCTION

1. UNIT STEP FUNCTION

3. Example: Solution:

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

5. Ex1: Ex2: Ex3: = ?

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

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

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

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

10. 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)}

11. Example:

12. Laplace transform of tf(t)

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