LaplaceTransform .

Transforms

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

LaplaceTransform Laplace Transform


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;