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