Semester 3MathematicsDifferential EquationsLaplace TransformLaplace DerivationsLaplace Frequency Shifting PropertyNote L{eatf(t)}=F(s−a)\mathcal{L}\{e^{at}f(t)\} = F(s - a)L{eatf(t)}=F(s−a) Derivation: L{f(t)}=∫0∞e−stf(t) dt=F(s)L{eatf(t)}=∫0∞e−steatf(t) dt=∫0∞e−st+atf(t) dt=∫0∞e−(s−a)tf(t) dt=F(s−a)\begin{align*} \mathcal{L}\{f(t)\} &= \int_{0}^{\infty} e^{-st}f(t)\, dt = F(s)\\ \mathcal{L}\{e^{at}f(t)\} &= \int_{0}^{\infty} e^{-st}e^{at}f(t)\, dt \\ &= \int_{0}^{\infty} e^{-st+at}f(t)\, dt \\ &= \int_{0}^{\infty} e^{-(s-a)t}f(t)\, dt \\ &= F(s-a) \end{align*} L{f(t)}L{eatf(t)}=∫0∞e−stf(t)dt=F(s)=∫0∞e−steatf(t)dt=∫0∞e−st+atf(t)dt=∫0∞e−(s−a)tf(t)dt=F(s−a)Solving Differential Equations with Laplace transformPrevious PageLaplace Time scaling PropertyNext Page