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      <text>20.3.5.1 The Laplace transform of the delta function</text>
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      <text><![CDATA[<h2>The Laplace transform of the delta function</h2>
<p>Here we consider \({\cal L}\{\delta(t-d)\}\). From the definition of the Laplace transform:
\[ {\cal L}\{\delta (t-d)\}=\int_0^{\infty}\text{e}^{-st}\delta (t-d)\,dt=\text{e}^{-sd} \]
by the sifting property of the delta function. Thus
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<h4>Key Point</h4>
<p><center><strong>Laplace Transform of the Delta Function</strong></center></p>
<p>\[{\cal L}\{\delta (t-d)\}=\text{e}^{-sd}\quad \text{ and,  putting  } \ d=0,\quad {\cal L}\{\delta (t)\}=\text{e}^0=1\]</p>
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<hr>
<h4 class="HELM_example">Example</h4>
<p>Find the Laplace transform of \(3\delta (t-3).\)</p>
<h4 class="HELM_solution">Solution</h4>
<p>\(3e^{-3s}\)</p>]]></text>
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