File: transforms.chapt.txt

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			Transforms

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

*CMD LaplaceTransform --- Laplace Transform
*STD
*CALL
	LaplaceTransform(t,s,f)	
*PARMS

{t} -- independent variable that is being transformed

{s} -- independent variable that is being transformed into

{f} -- function

*DESC

This function attempts to take the function $f(t)$ and find the Laplace transform
of it,$F(s)$, which is defined as $Integrate(t,0,Infinity) Exp(-s*t)*f(t)$. 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.


*E.G.

	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;