Question

laplace transform of unit step function
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Answer 1

Answer:

laplace transform of unit step function:

Recall \displaystyle{u}{\left({t}\right)}u(t) is the unit-step function.

1. ℒ\displaystyle{\left\lbrace{u}{\left({t}\right)}\right\rbrace}=\frac{1}{{s}}{u(t)}=

s

1

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2. ℒ\displaystyle{\left\lbrace{u}{\left({t}-{a}\right)}\right\rbrace}=\frac{{e}^{{-{a}{s}}}}{{s}}{u(t−a)}=

s

e

−as

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3. Time Displacement Theorem:

If \displaystyle{F}{\left({s}\right)}=F(s)= ℒ\displaystyle{\left\lbrace f{{\left({t}\right)}}\right\rbrace}{f(t)} then ℒ\displaystyle{\left\lbrace{u}{\left({t}-{a}\right)}\cdot g{{\left({t}-{a}\right)}}\right\rbrace}={e}^{{-{a}{s}}}{G}{\left({s}\right)}{u(t−a)⋅g(t−a)}=e

−as

G(s)