Yes your teacher is right. The formula for the laplace transform of U_a f(t-a) is e^-as F(s).

To use this formula, we have to transform the given function to the form Ua f(t-a).

The given function is f(t) = g(t) U (t-a)

= g(t) U_t (a) … because U_t a= U(t-a)

=g(t+a –a) U_t (a) Or U_t (a) g (t+a-a)

Now we can apply the formula. The Laplace transform would be e^-as L g(t+a)

Consider the following example, which would make it easier to understand, how to apply the formula.

Let f(t) = Cos t U (t-5) Or, Cos t U₅ (t). To conform to the formula given above , write

f(t) = Cos (t+5-5) U₅ (t). Laplace transform of f(t) would be e^-5s{ L Cos (t+5)}

Or e^-5s {L (cos 5 Cost-Sin 5 Sint)}

Or e^-5s {Cos5 . s/(s² +1) – Sin5 . 1/(s² +1)}

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