kill(all)$

representation : dirac-delta function ,version 1.0
peter.vlasschaert@gmail.com,10/02/2020

load packages
===========

load(lrats);

representation
===========

assume(a>0);

very important integral
================
analyse : complex variable , 'complex contour'

p1:'integrate(exp(-a*x^2+b*x), x, minf, inf) =integrate(exp(-a*x^2+b*x), x, minf, inf) ;

function : dirac delta function.

sin(x)/x ?

p2:'integrate(sin(x)/x,x,minf,inf) =integrate(sin(x)/x,x,minf,inf);

p2a:p2/π;

limit  x,  0 ,∞,-∞
============

p3:'limit(sin(x)/x,x,0) =limit(sin(x)/x,x,0) ;

sin(0);

0/0  ? 

result :  ∞/∞ ,0/0  = >  L'Hopital rule 

p3a: 'limit(sin(x)/x,x,inf) =limit(sin(x)/x,x,inf) ;
p3b: 'limit(sin(x)/x,x,minf) =limit(sin(x)/x,x,minf) ;

rem :  x → π*x

'limit(sin(x*π)/(x*π),x,0) =limit(sin(x*π)/(x*π),x,0) ;

p3a: 'limit(sin(x*π)/(x*π),x,inf) =limit(sin(x*π)/(x*π),x,inf) ;
p3b: 'limit(sin(x*π)/(x*π),x,minf) =limit(sin(x*π)/(x*π),x,minf) ;

p3c:wxplot2d([sin(x)/x], [x,-100,100])$

better : representation dirac-delta.
-----------------------------------------

p3d:subst([x=π*x],sin(x)/x);
p3d1:wxplot2d([p3d], [x,-100,100])$

normal distribution :σ ' = standard deviation '
================================

assume(σ>0);

p4:1/(sqrt(2*π)*σ)*exp((-x^2)/(2*σ^2));

p4a:p4*sqrt(2*π)*σ;

p4a1:denom(p4);

p4b:'integrate(p4a,x,minf,inf) =integrate(p4a,x,minf,inf);

propertie : dirac -delta function.
=======================

p4b1:p4b/p4a1;

limit  σ → 0,x = 0
=============
peak = to infinity

p5:subst([x=0],p4);

p5a:'limit(p5,σ,0) = limit(p5,σ,0);

limit  σ > 0, x = -∞,∞
===============

p5b1:'limit(p4,x,minf) = limit(p4,x,minf);
p5b2:'limit(p4,x,inf) = limit(p4,x,inf);

plot
σ≠ value,normal distribution
====================

p6a1:normf(σ):=''p4;

p6a2:wxplot2d( [normf(10),normf(1),normf(0.1)],[x,-30,30]);

ideal representation : dirac-delta function
==============================
normal distribution 'p4' :  x ≠ 0, σ  → 0

p6a3:delta(x) = 'limit(p4,σ,0);

rem : dirac-delta , function not implimented in general 'Maxima',
only for Laplace Transformations.

properties
=======

1e) general
--------------

p7:'integrate(f(x)*delta(x-c),x,minf,inf) =f(c);

2e) c >>>ε very small 
---------------------------

p7a:'integrate(f(x)*delta(x-c),x,-c-ε,c+ε) =f(c);

  =================================

       by definition :delta(x-c) ?
       ---------------------------------
  =>  delta(0)   , only when x= c
       ---------------------------------
       general :ε  → 0, 'p7a'

 =================================

fourier pair (f(x),F(k))
================
k : frequency domain
x 

p8a:f(x) =1/sqrt(2*π)*integrate(F(k)*exp(%i*x*k),k,minf,inf);
p8b:F(k) =1/sqrt(2*π)*integrate(f(x)*exp(-%i*x*k),x,minf,inf);

a) subst:F(k) → f(x)
------------------------
F(k) = >  x→xx

p8b1:subst([x=xx],p8b);

b) subst:'p8b1' → 'p8a'
----------------------------

p8b2:subst([p8b1],p8a);

p8b3:args(rhs(p8b2));

principal :p7
======

first : integral dirac-delta function
----------------------------------------

p8b4:delta(x-xx) = 1/p8b3[2]*integrate(exp(%i*k*(x-xx)),k,minf,inf);

second : integral dirac-delta function
---------------------------------------------

p8b4a:subst([%i=-%i],p8b4);

rem :complex
==========

p8b4b:(%i)^2;

rhs(p8b4)
========

p8b41:rhs(p8b4);

rem:

p7:'integrate(f(x)*delta(x-xx),x,minf,inf) =f(xx);

  How to use: changevar
  =================

p8b5:'integrate(sin(2*x),x)=integrate(sin(2*x),x);

p8b51:changevar(lhs(p8b5), y=2*x, y, x);

p8b52:ev(p8b51,integrate);

p853:subst([y=2*x],p8b52);

  ==================

sin(g*x)/(π*x)
==========
g → ⇔ = inf

assume(g>0);

q(g):=sin(g*x)/(π*x);

p9:'integrate(q(g),x,minf,inf)  = integrate(q(g),x,minf,inf);

p9a:'limit(q(g),x,0) = limit(q(g),x,0);

p9a1:'limit(lhs(p9a),g,inf) = limit(rhs(p9a),g,inf);

defintion :dirac-delta function.
======================

p9a2:delta(x) = 'limit(q(g),g,inf);

p9a2a:rhs(p9a2) = lhs(p9a2);

how to eval : integral :exp(%i*k*x)  
=========================
see also:p8b4, k = variable

p10a:1/(2*π)*'integrate(exp(%i*k*x),k,-g,g)$
p10b:combine(1/(2*π)*integrate(exp(%i*k*x),k,-g,g))$

p10:p10a=p10b;

exp(%i*θ) = ?
-----------------
θ = g*x

p10c:sin(θ);

p10c1:exponentialize(p10c);

p11:subst([θ=g*x],p10c1=p10c);

p12:lhs(p10) =lratsubst([p11],rhs(p10));

p12a:'limit(lhs(p12),g,inf) ='limit(rhs(p12),g,inf);

see :"p9a2a"

p12b:subst([p9a2a],p12a);

p12b1:subst([x=x-x0],rhs(p12b) = lhs(p12b));

relation : between , fourier pairs
========================
use : p7,x0 integration variable

p13:f(x) =integrate(delta(x-x0)*f(x0),x0,minf,inf);

p13a:subst([p12b1],p13);

F(k) iintegration over x0
-----------------------------

p14:F(k) =1/(sqrt(π*2))*integrate(exp(-%i*k*x0)*f(x0),x0,minf,inf);

p14a:subst([x0=x],p14);

f(x) iintegration over k
-----------------------------

p14b:f(x)=1/(sqrt(π*2))*'integrate(exp(%i*k*x)*F(k),k,minf,inf);

rem : fourier transform : p14a

find : fourier transform , distribution
==========================

1e) distribution
------------------

p15a:f(x) =  A*exp(-x^2/(2*σ^2));

p15a1:'limit(rhs(p15a),x,σ) =limit(rhs(p15a),x,σ);

2e) find fourier transform
-------------------------------

p15b:subst([p15a],p14a);

see : also "p1"

p15b1:ev(p15b,nouns);

p15b2:'limit(rhs(p15b1),k,1/σ) =limit(rhs(p15b1),k,1/σ);

 =======================================

rem :
f(x)  → σ    → Δx
F(k) → 1/σ → Δk