Laplace Transform

Author

Eric W. Weisstein
October 4, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegralTransforms/LaplaceTransform.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LaplaceTransform.html.

©2007 Wolfram Research, Inc. except for portions noted otherwise

Table

fns={1,t,t^n,t^a,Exp[a t],Cos[ω t],Sin[ω t],Cosh[ω t],Sinh[ω t],Exp[a t]Sin[b t],Exp[a t]Cos[b t],DiracDelta[t-c],UnitStep[t-c],BesselJ[0,t],BesselJ[n,t]};

{#,FullSimplify[LaplaceTransform[#,t,s,Assumptions->n∈Integers]]}&/@fns//TableForm//TraditionalForm

{#,l=LaplaceTransform[#,t,s,GenerateConditions->True,Assumptions->n∈Integers];Sequence@@If[Head[l]===If,{FullSimplify[l[[2]]],l[[1]]},{FullSimplify[l],True}]}&/@fns//TableForm//TraditionalForm

Special cases

(Subscript[ℒ, t])[(t^a)](s)

LaplaceTransform[t^a,t,s,GenerateConditions->True]

(Subscript[ℒ, t])[θ(t-c)](s)

v4.2

LaplaceTransform[UnitStep[t-c],t,s]

v5.0

LaplaceTransform[UnitStep[t-c],t,s]

v6

LaplaceTransform[UnitStep[t-c],t,s]

(Subscript[ℒ, t])[(Subscript[J, 0])(a t)](s)

LaplaceTransform[BesselJ[0,a t],t,s]

FullSimplify[%,s>0]

(Subscript[ℒ, t])[(Subscript[J, n])(a t)](s)

LaplaceTransform[BesselJ[n,a t], t,s]

FullSimplify[FunctionExpand[%]]

FullSimplify[%,s>0]

Compare with A&S, p. 1024, 29.3.56

FullSimplify[((((((a^2)))^(n/2))(((s+(Sqrt[(a^2)+(s^2)])))^(-n)))/(Sqrt[(a^2)+(s^2)]))-(((Sqrt[s^2+a^2]-s)^n)/((a^n)Sqrt[s^2+a^2]))]

FullSimplify[-(a^(-n)) (((-s+(Sqrt[(a^2)+(s^2)])))^n)+((((a^2)))^(n/2)) (((s+(Sqrt[(a^2)+(s^2)])))^(-n)),n∈Integers&&a>0]

FullSimplify[(a^(2n))(((s+(Sqrt[(a^2)+(s^2)])))^(-n))-(Sqrt[s^2+a^2]-s)^n,n∈Integers]

Derivatives

f'(t)

LaplaceTransform[f'[t],t,s]

f''(t)

LaplaceTransform[f''[t],t,s]

Heaviside Calculus

Solve[LaplaceTransform[f''[t]+a1 f'[t]+a0 f[t],t,s]==0,LaplaceTransform[f[t],t,s]
]

InverseLaplaceTransform[((a1 f[0]+s f[0]+f'[0])/(a0+a1 s+(s^2))),s,t]

FullSimplify[%]

g[t_]:=(1/(Sqrt[-4 a0+(a1^2)]))((E^(-((a1 t)/2))) ((Sqrt[-4 a0+(a1^2)]) Cosh[(1/2) (Sqrt[-4 a0+(a1^2)]) t] f[0]+Sinh[(1/2) (Sqrt[-4 a0+(a1^2)]) t] (a1 f[0]+2 f'[0])))

Simplify[g''[t]+a1 g'[t]+a0 g[t]]

DSolve[f''[t]+a1 f'[t]+a0 f[t]==0,f[t],t]

Transformations

Inverse

Solve[F[s]==LaplaceTransform[f[t],t,s],f[t]]

Shift

FullSimplify[{LaplaceTransform[#,t,s],LaplaceTransform[Exp[a t]#,t,s]}]&/@fns//ColumnForm//TraditionalForm

Integrals

FullSimplify[{LaplaceTransform[#,t,s]/s,LaplaceTransform[Integrate[#,{t,0,t}],t,s]}]&/@fns//ColumnForm