Table of the Laplace Transforms
Related Calculators: Laplace Transform Calculator , Inverse Laplace Transform Calculator
This is not complete list of Laplace Transforms, but it contains all common transforms:
| ` f(t)=L^(-1)(F(s)) ` | `F(s)=L(f(t))` |
| 1 | `1/s` |
| `t^n`, n=0, 1, 2, 3... | `(n!)/(s^(n+1))` |
| `t^n`, n>-1 | ` (Gamma(n+1))/s^(n+1) ` |
| `e^(at)` | `1/(s-a)` |
| `t^(n-1/2)`, n=1,2,3.. | `(1*3*5*...*(2n-1)*sqrt(pi))/(2^ns^(n+1/2))` |
| `sqrt(t)` | `sqrt(pi)/(2s^(3/2))` |
| `sin(at)` | `a/(s^2+a^2)` |
| `cos(at)` | `s/(s^2+a^2)` |
| `sinh(at)` | `a/(s^2-a^2)` |
| `cosh(at)` | `s/(s^2-a^2)` |
| `tsin(at)` | `(2as)/(s^2+a^2)^2` |
| `tcos(at)` | `(s^2-a^2)/(s^2+a^2)^2` |
| `sin(at+b)` | `(s*sin(b)+a*cos(b))/(s^2+a^2)` |
| `cos(at+b)` | `(s*cos(b)-a*sin(b))/(s^2+a^2)` |
| `e^(at)sin(bt)` | `b/((s-a)^2+b^2)` |
| `e^(at)cos(bt)` | `(s-a)/((s-a)^2+b^2)` |
| `e^(at)sinh(bt)` | `b/((s-a)^2-b^2)` |
| `e^(at)cosh(bt)` | `(s-a)/((s-a)^2-b^2)` |
| `t^n e^(at)`, n=1,2,3... | `(n!)/(s-a)^(n+1)` |
| `f(ct)` | `1/cF(s/c)` |
| `u_c(t)=u(t-c)` | `e^(-cs)/s` |
| `u_c(t)f(t-c)` | `e^(-cs)F(s)` |
| `\delta(t-c)` | `e^(-cs)` |
| `\e^(ct)f(t)` | `F(s-c)` |
| `t^nf(t)`, n=1,2,3... | `(-1)^nF^((n))(s)` |
| `int_0^tf(\tau)d\tau` | `(F(s))/s` |
| `int_0^tf(t-tau)g(tau)dtau` | `F(s)G(s)` |
| `f'(t)` | `sF(s)-f(0)` |
| `f''(t)` | `s^2F(s)-s f(0)-f'(0)` |
| `f^((n))(t)` | `s^nF(s)-sum_(k=0)^(n-1)(s^(n-1-k)f^((k))(0))` |