## File: transforms.chapt.txt

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yacas 1.3.6-2
 `123456789101112131415161718192021222324252627282930313233343536373839404142` `````` 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; ``````