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))`