Laplace Transform: Some Proofs

 

1. If \( F(s)=\mathcal{L}\{ f(t)\}\) exists for \(s>a\ge 0\), and \(c\) is a constant, then prove  \(\mathcal{L}\{u(t-c)\}=\dfrac{e^{-cs}}{s}.\)   Proof

 

2. If \( F(s)=\mathcal{L}\{ f(t)\}\) exists for \(s>c\ge 0\), and \(a\) is a constant, then prove  \(\mathcal{L}\{e^{at}f(t)\}=F(s-a).\)  Proof

 

3. If \( F(s)=\mathcal{L}\{ f(t)\}\) exists for \(s>c\ge 0\), and \(a\) is a constant, then prove  \(\mathcal{L}\{ f(t-a) u(t-a)\}=e^{-as}F(s).\)   Proof

 

4. If \( F(s)=\mathcal{L}\{ f(t)\}\) exists for \(s>a\ge 0\), and \(c\) is a constant, then prove  \(\mathcal{L}\{ f(ct)\}=\frac{1}{c}\,F\left(\frac{s}{c}\right).\)   Proof