Laplace Transform Table
Formula1
This is table of Laplace transform.
| $f(t)=\mathcal{L^{-1}}$ | $F(s)=\mathcal{L} \left\{ f(t) \right\}$ | Derivation |
|---|---|---|
| $1$ | $\dfrac{1}{s}$ | link |
| $e^{at}$ | $\dfrac{1}{s-a}$ | link |
| $t^n$ | $\dfrac{n!}{s^{n+1}}$ | link |
| $t^{p}$ | $\dfrac{ \Gamma (p+1) }{ s^{p+1}}$ | link |
| $t^{p}e^{at}$ | $\dfrac{ \Gamma (p+1) }{ (s-a)^{p+1}}$ | link |
| $\sin (at)$ | $\dfrac{a}{s^2+a^2}$ | link |
| $\cos (at)$ | $\dfrac{s}{s^2+a^2}$ | link |
| $e^{at}\sin(bt)$ | $\dfrac{b}{(s-a)^2 +b^2}$ | link |
| $e^{at}\cos(bt)$ | $\dfrac{s-a}{(s-a)^2+b^2}$ | link |
| $\sinh (at)$ | $\dfrac{a}{s^2-a^2}$ | link |
| $\cosh (at)$ | $\dfrac{s}{s^2-a^2}$ | link |
| $e^{at} \sinh (bt)$ | $\dfrac{b}{(s-a)^2-b^2}$ | link |
| $e^{at} \cosh (bt)$ | $\dfrac{s-a}{(s-a)^2-b^2}$ | link |
| $u_{c}(t)= \begin{cases} 0 & t<c \\ 1 & t\ge c\end{cases}$ | $\dfrac{e^{-cs}}{s}$ | link |
| $u_{c}(t)f(t-c)$ | $e^{-cs}F(s)$ | link |
| $f^{\prime}(t)$ | $s\mathcal{L} \left\{ f(t) \right\} -f(0)$ | link |
| $f^{(n)}$ | ${s^n\mathcal {L}\left\{ f(t) \right\} -s^{n-1}f(0) - \cdots -f^{(n-1)}(0) }$ | link |
| $f(t)=f(t+T)$ | $\dfrac{\displaystyle \int_{0}^T e^{-st}f(t)dt}{1-e^{-st}}$ | link |
| $\delta (t-t_{0})$ | $e^{-st_{0}}$ | link |
| $f(ct)$ | $\frac{1}{c}F \left( \frac{s}{c} \right)$ | link |
| $\frac{1}{k}f (\frac{t}{k} )$ | $F(ks)$ | link |
| $\frac{1}{a} e^{-\frac{b} {a}t}f\left(\frac{t}{a}\right)$ | $F(as+b)$ | link |
| $t^{n}f(t)$ | $(-1)^{n}F^{(n)}(s)$ | link |
William E. Boyce , Boyce’s Elementary Differential Equations and Boundary Value Problems (11th Edition, 2017), Chapter6 The Laplace Transform ↩︎