kill(all)$

load(lrats);

relation between : inverse laplace transform & fourier transform,version 1.0
peter.vlasschaert@gmail.com,04/08/2019

Laplace transform
================

def: laplace transform
------------------------------

assume
f(t) = 0 → t < 0
f(t)≠ 0  → t>0

p1:F(s)  = 'integrate(f(t)*exp(-s*t),t,0,∞);

use: s=γ+i*ω   (  complex value : z= x+iy,gauss plane)

p2:s=γ+%i*ω; 

p2a:rhs(p2) = lhs(p2);

p2b:t*p2a;

subst; p2 into p1

p3:lhs(p1) =subst([p2],rhs(p1));

Given : F(s)  => f(t)

a) Fourier transform
===============

def : Fourier transform ,'f(t)'
-------------------------------------

p4a:F(ω) = integrate(f(t)*exp(-%i*ω*t),t,-∞,∞);;

def : inverse Fourier transform: 'p4a'
--------------------------------------------------

p4b:f(t) = 1/(2*π)* integrate(F(ω)*exp(%i*ω*t),ω,-∞,∞);;

b) φ(t) = f(t)*exp(-γ*t)
================
assume
φ(t) = 0                     → t < 0
φ(t) =  f(t)*exp(-γ*t)  → t>0
rem :γ = constant

p4c:φ(t) =  f(t)*exp(-γ*t) ;

1e)  fourier transform.
================

p5:ψ(ω)=integrate(φ(t)*exp(-%i*ω*t),t,-∞,∞);

p5a:args(rhs(p5));

p5b:subst([p4c],p5a[1]);

p5c:subst([f(t) =1],p5b);

use : trick (can't use factor,gfactor)

p5d:exp(-h*t);

find : h
----------

p5e:solve(p5c=p5d,h)[1];

integrand
--------------

p5f:f(t)*subst([p5e],p5d);

p5f1:subst([p2a],p5f);

integration
---------------
use : definition of the laplace.

p5g:lhs(p5) =rhs(p1);

p5g1:lhs(p5g) =lhs(p1);

p5g2:lhs(p5g1) = subst([p2],rhs(p5g1));

2e)  inverse fourier transform.
======================

p6:φ(t) = 1/(2*π)* integrate(ψ(ω)*exp(%i*ω*t),ω,-∞,∞);;

p6a:args(p6);

p6b:args(p6a[2]);

p6c:p6b[1];

representation integral :p6*2*π
========================

p6d:args(p6c);

p6e:(p6d[1]*exp(γ*t));

p6f:subst([p5g2], p6e);

p6f1:exp(s*t)*subst([p2a],part(p6f,2));

p6g:f(t)=1/p6b[2]*'integrate(p6f,ω,-∞,∞);

ω  to s
=====

p7a:p2;

p7b:diff(p7a);

insert : because γ = constant

p7c:subst([del(γ) =0],p7b);

del(ω)=del(s)/(%i);

integration : f(t) with respect to s
---------------------------------------------

p8:[p6f1,s,γ+%i*ω1,γ+%i*ω2];

p8a:lratsubst([ω1=p6d[3],ω2=p6d[4]],p8);

p8b:1/(p6b[2]*%i)*'integrate(p8a[1],p8a[2],p8a[3],p8a[4]);

=>   -1= (%i)^2, i= %i

def : inverse laplace transform.
=======================

p8c:lhs(p6g)=1/(2*π*i) *'integrate(F(s)*%e^(s*t),s,γ-%i*inf,γ+%i*inf);