Discussion
Laplace transform of \(\delta(t - T)\) is :
Answer: C
The Laplace transform of \(\delta(t - T)\) is \(e^{-sT}\).
Using the sifting property of the impulse, for \(T \gt 0\):
\[ \int_{0}^{\infty} \delta(t - T) e^{-st}\, dt = e^{-sT} \]
This is also the time-shift theorem applied to \(\mathcal{L}\{\delta(t)\} = 1\): delaying by T multiplies the transform by \(e^{-sT}\).
'Constant' is the transform of the undelayed impulse \(\delta(t)\). \(1/e^{-sT} = e^{sT}\) would be an advance, not a delay, and \(s e^{-sT}\) is the transform of the derivative of a delayed impulse.
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