Exercise · ZW-LAPLACE-TRANSFORM-009 · ★★★ Advanced
For which values of \(s\) does the integral defining \(\mathcal L\{e^{2t}\}(s)\) converge?
Choose one option.
Show answer
- B. \(s>2\)
Explanation
The integrand is \(e^{-(s-2)t}\), which decays only if \(s-2>0\). For \(s\le2\) the integral diverges.
Review the concept: Laplace transform
The Laplace transform \(\mathcal L\) assigns to a function \(f(t)\), defined for \(t\ge0\), the function \(F(s)\) obtained by integrating \(f\) times \(e^{-st}\) over \([0,\infty)\).
\[\mathcal L\{f\}(s)=F(s)=\int_0^\infty e^{-st}f(t)\,dt\]
More practice problems: laplace transform
- ZW-LAPLACE-TRANSFORM-001 Compute mathcal L\1\(s) for s>0.
- ZW-LAPLACE-TRANSFORM-002 Compute mathcal L\t\(s).
- ZW-LAPLACE-TRANSFORM-003 Compute mathcal L\e^3t\(s) for s>3.
- ZW-LAPLACE-TRANSFORM-004 Compute mathcal L\sin 3t\(s).
- ZW-LAPLACE-TRANSFORM-005 Compute mathcal L\cos 2t\(s).
- ZW-LAPLACE-TRANSFORM-006 Compute mathcal L\t^2\(s).
- ZW-LAPLACE-TRANSFORM-007 Use linearity to compute mathcal L\3+2t\(s).
- ZW-LAPLACE-TRANSFORM-008 Compute mathcal L\te^t\(s).