Laplace Transforms

Here you will find the Laplace transform of functions as well as their inverse Laplace transform. The Laplace transform table, formula, properties, and applications to find the solution of an ODE will be discussed here.

Main Article: Laplace Transform: Definition, Table, Formulas, Properties

Change of Scale Property

First Shifting Property: Formula, Proof

Second Shifting Property: Formula, Proof

Laplace Transform of Periodic Functions

Laplace Transform of Derivatives

Laplace Transform of Integrals

Laplace transform of unit step function

Laplace of 1/t does NOT exist: Proof

For the Laplace transforms of your desired functions, just click on the function, and we will get your answer.

Laplace transforms of Basic Functions:

Functions f(t) L{f(t)}
1L{1} = 1/s
tL{t} = 1/s2
t2L{t2} = 2/s3
t3L{t3} = 6/s4
tnL{tn} = n!/s^{n+1}
2tL{2t} =1/(s-ln2), s>ln2
eatL{eat} = 1/(s-a)
sintL{sint} = 1/(s2+1)
sinatL{sin at} = a/(s2+a2)
costL{cost} = s/(s2+1)
cosatL{cos at} = s/(s2+a2)

More Laplace transforms:

tett sintt cost
teatt sinatt cosat
sint costcost cos2t
sin2t sin3tsint sin2t sin3tt2sin2t
t sin2tsin3t(1-sint)/t
sinh(at)cosh(at)t sinht
t cosht
e-t sintt2cosat

Laplace of Unit Step Functions:


Laplace transform of (eat-cosbt)/t

Inverse Laplace Transforms

Table of Inverse Laplace Transformations

Inverse Laplace transform of 1

Inverse Laplace transform of 1/s

Inverse Laplace of 1/s2

Inverse Laplace of 1/s3

Inverse Laplace of 1/s(s+1)

Inverse Laplace of s/s+1