Laplace Transform of Heaviside Step Function times Function

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Theorem

Let $\map f t: \R \to \R$ or $\R \to \C$ be a function of exponential order $a$ for some constant $a \in \R$.

Let $f$ be piecewise continuous with one-sided limits on any closed interval of the form $\closedint 0 b$ where $b > 0$.

Let $\map {u_c} t$ be the Heaviside step function.

Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$.


Then:

$\laptrans {\map {u_c} t \, \map f {t - c} } = e^{-s c} \map F s$

for $\map \Re s > a$.


Proof

\(\displaystyle \laptrans {\map {u_c} t \, \map f {t - c} }\) \(=\) \(\displaystyle \int_0^{\to +\infty} \map {u_c} t e^{-s t} \map f {t - c} \rd t\) $\quad$ Definition of Laplace Transform $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int_0^{\to c^-} \map {u_c} t e^{-s t} \map f {t - c} \rd t + \int_{\to c^+}^{\to +\infty} \map {u_c} t e^{-s t} \map f {t - c} \rd t\) $\quad$ Sum of Complex Integrals on Adjacent Intervals $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int_0^{\to c^-} 0 \times e^{-s t} \map f {t - c} \rd t + \int_{\to c^+}^{\to +\infty} 1 \times e^{-s t} \map f {t - c} \rd t\) $\quad$ Definition of Heaviside Step Function $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int_{\to c^+}^{\to +\infty} e^{-s t} \map f {t - c} \rd t\) $\quad$ $\quad$

Let $u = t - c$.

Then $\dfrac {\rd u} {\rd t} = 1$.

Then $u \to 0^-$ as $t \to c^+$.

Also, $u \to +\infty$ as $t \to +\infty$.

So:

\(\displaystyle \int_{\to c^+}^{\to +\infty} e^{-s t} \map f {t - c} \rd t\) \(=\) \(\displaystyle \int_{\to c^+}^{\to +\infty} e^{-s \paren {u + c} } \map f u \frac {\rd u} {\rd t} \rd t\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int_{\to 0^-}^{\to +\infty} e^{-s \paren {u + c} } \map f u \rd u\) $\quad$ Integration by Substitution $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int_0^{\to 0^+} e^{-s \paren {u + c} } \map f u \rd u + \int_{\to 0^-}^{\to +\infty} e^{-s \paren {u + c} } \map f u \rd u\) $\quad$ Integral on Zero Interval $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int_0^{\to +\infty} e^{-s \paren {u + c} } \map f u \rd u\) $\quad$ Sum of Complex Integrals on Adjacent Intervals $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int_0^{\to +\infty} e^{-s u - s c} \map f u \rd u\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle e^{-s c} \int_0^{\to +\infty} e^{-s u} \map f u \rd u\) $\quad$ Exponent Combination Laws $\quad$
\(\displaystyle \) \(=\) \(\displaystyle e^{-s c} \int_0^{\to +\infty} e^{-s t} \map f t \rd t\) $\quad$ renaming the variable of integration $\quad$
\(\displaystyle \) \(=\) \(\displaystyle e^{-s c} \laptrans {\map f t}\) $\quad$ Definition of Laplace Transform $\quad$

$\blacksquare$


Sources