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inttrans

  

laplace

  

Laplace transform

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

laplace(expr, t, s)

Parameters

expr

-

expression, equation, or set of expressions and/or equations to be transformed

t

-

variable expr is transformed with respect to t

s

-

parameter of transform

opt

-

option to run this under (optional)

Description

• 

The laplace function computes the Laplace transform (F(s)) of expr (f(t)) with respect to t, using the definition:

Fs=0ftⅇstⅆt

• 

Expressions involving a wide variety of functions including exponentials, trigonometrics, Bessel functions, error functions, and many others can be transformed.

• 

The laplace function also recognizes derivatives (diff or Diff) and integrals (int or Int).

• 

When transforming expressions like diff(y(t), t, s), laplace will insert the initial values y0, Dy0, etc.   Dy0 is the value of the first derivative at 0; DDy0 is the second derivative at 0, and so on.

• 

Both laplace and invlaplace recognize the Dirac-delta (or unit-impulse) function as Dirac(t) and Heaviside's unit step function as Heaviside(t).

• 

Users can add their own functions to laplace's internal lookup table by using the addtable function.

• 

If the option opt is set to 'NO_INT', then the program will not resort to integration of the original problem if all other methods fail.  This will increase the speed at which the transform will run.

• 

The command with(inttrans,laplace) allows the use of the abbreviated form of this command.

Examples

withinttrans:

laplacet2+sint=yt,t,s

2s3+1s2+1=laplaceyt,t,s

(1)

laplacet32ⅇt+sinhat,t,s

34πs5/21s1+aa2+s2

(2)

laplaceⅆ2ⅆt2ytyt=sinat,t,s2

s22laplaceyt,t,s2Dy0s2y0laplaceyt,t,s2=as22+a2

(3)

laplaceBesselI0,at,t,s

1a2+s2

(4)

laplaceHeavisidetcft,t,s

laplaceHeavisidetcft,t,s

(5)

assumec,positive

laplaceHeavisidetcft,t,s

ⅇsc~laplaceft+c~,t,s

(6)

addtablelaplace,myfunct,Myfuncs,t,s:

laplacet3ⅇatmyfunc4t,t,w

1256D3Myfunc14w14a

(7)

addtablelaplace,myfunc2tan,1Myfunc2sn+12!+a,t,s,a,n,n::odd:

laplacemyfunc24t7,t,w

124Myfunc2w+4

(8)

See Also

dsolve

inttrans

inttrans[addtable]

inttrans[invlaplace]

 


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