kill(all)$

maxima : appendix: integral transform : Laplace Transform ,version 1.0
peter.vlasschaert@gmail.com, 31/05/2022

appendix : Laplace Transform.
======================

definition : integraltransform.(= p1)
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p1:integrate(g(t)*k(x,t),t,0,inf) = G(x);

set : k(x,t) = x^t

p2:subst([k(x,t)=x^t],p1);

→ x = e^w, find w

rem : maxima : log(x) = ln(x) in math

p3:x = exp(w);

p3a:log(p3);

p3a1:reverse(p3a);

→ subst : p3a1 into p3

simp:false;

p3b:lhs(p3)=subst([p3a1],rhs(p3));

→ x^t

p3b1:(lhs(p3))^t =(rhs(p3b))^t;

when : t→ ∞ then (x< 1) => ?<x<1

simp:true;

p4a:log(1);

p4b:wxplot2d(log(x),[x,0,5]);

from graph : t → ∞ ; x : 0<x<1 

 'limit (log(v), v, 0, plus) = limit (log(v), v, 0, plus);

reason : x : 0 < x ≤ 1

 'limit (log(v), v, 0, minus) = limit (log(v), v, 0, minus);

  limit (log(v), v, 0, plus)=gruntz (log(v), v, 0, plus);

rem :Dominik Gruntz, "On Computing Limits in a Symbolic Manipulation System" (ETH Zurich, 1996)"

p4:log(x) < 0;

p4a:-log(x)>0;

set : log(x) =-s

assume(s>0);

p5:log(x)=-s;

simp:false;

use : p3b1

p6:subst([p5],p3b1);

simp:true;

p6a:ratsimp(p6);

p6b:subst([t=1],p6a);

final result : F(s)
_________ 

→ use :p2

p7:subst([p6b],p2);

definition : Laplace Transform
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ref :https://en.wikipedia.org/wiki/Laplace_transform

p8:L(f(t))= F(s); /*see literature*/
p8a:rhs(p8)=integrate(exp(-s*t)*f(t),t,0,inf);

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