# 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))