LaplaceTransform .

# Transforms

In this chapter, some facilities for various transforms are described.

LaplaceTransform | Laplace Transform

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LaplaceTransform -- Laplace Transform

##### Standard library

##### Calling format:

    LaplaceTransform(t,s,func)

##### Parameters:

`t` -- independent variable that is being transformed

`s` -- independent variable that is being transformed into

`f` -- function

##### Description:

This function attempts to take the function `f(t)` and find the Laplace transform of it,`F(s)`, which is defined as `Integrate(t,0,Infinity) Exp(-s*t)*f`. This is also sometimes referred to the "unilateral" Laplace tranform. `LaplaceTransform` can transform most elementary functions that do not require a convolution integral, as well as any polynomial times an elementary function. If a transform cannot be found then `LaplaceTransform` will return unevaluated. This can happen for function which are not of "exponential order", which means that they grow faster than exponential functions.

##### Examples:

    In> LaplaceTransform(t,s,2*t^5+ t^2/2 )
    Out> 240/s^6+2/(2*s^3);
    In> LaplaceTransform(t,s,t*Sin(2*t)*Exp(-3*t) )
    Out> (2*(s+3))/(2*(2*(((s+3)/2)^2+1))^2);
    In> LaplaceTransform(t,s, BesselJ(3,2*t) )
    Out> (Sqrt((s/2)^2+1)-s/2)^3/(2*Sqrt((s/2)^2+1));
    In> LaplaceTransform(t,s,Exp(t^2)); // not of exponential order
    Out> LaplaceTransform(t,s,Exp(t^2));
    In> LaplaceTransform(p,q,Ln(p))
    Out> -(gamma+Ln(q))/q;