Laplace Transform of Dirac Delta Function by Function
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Theorem
Let $\map f t: \R \to \R$ or $\R \to \C$ be a function.
Let $\map \delta t$ denote the Dirac delta function.
Let $c$ be a positive constant real number.
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$.
Then:
- $\laptrans {\map \delta {t - c} \map f t} = e^{- s c} \map f c$
Proof
\(\ds \laptrans {\map \delta {t - c} \map f t}\) | \(=\) | \(\ds \int^{\to+\infty}_0 e^{-s t} \map \delta {t - c} \map f t \rd t\) | Definition of Laplace Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds \int^{c^+}_{c^-} e^{-s t} \map \delta {t - c} \map f t \rd t\) | Integrand elsewhere zero by Definition of Dirac Delta Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \int^{c^+}_{c^-} e^{-s c} \map \delta {t - c} \map f c \rd t\) | $e^{-s t}$ and $\map f t$ are constant in interval $\closedint {c^-} {c^+}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{-s c} \map f c \int^{c^+}_{c^-} \map \delta {t - c} \rd t\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{-s c} \map f c \int^{0^+}_{0^-} \map \delta {t - c} \map \rd {t - c}\) | Integration by Substitution | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{-s c} \map f c\) | Definition of Dirac Delta Function |
$\blacksquare$