reduce-historical/r36/xlog/LAPLACE.LOG

REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...


% Title:  Examples of Laplace Transforms.

% Author: L. Kazasov.

% Date: 24 October 1988.

order p;



% Elementary functions with argument k*x, where x is object var.

laplace(1,x,p);


 1
---
 p

laplace(c,x,p);


 c
---
 p

laplace(sin(k*x),x,p);


    k
---------
  2    2
 p  + k
 laplace(sin(x/a),x,p);


        1
-----------------
  -1   2  2
 a  *(p *a  + 1)

laplace(sin(17*x),x,p);


    17
----------
  2
 p  + 289

laplace(sinh x,x,p);


   1
--------
  2
 p  - 1

laplace(cosh(k*x),x,p);


     - p
------------
     2    2
  - p  + k

laplace(x,x,p);


 1
----
  2
 p
 laplace(x**3,x,p);


 6
----
  4
 p

off mcd;

 laplace(e**(c*x) + a**x, x, s);


                -1          -1
 - ((log(a) - s)   + (c - s)  )

laplace(e**x - e**(a*x) + x**2, x, p);


   -3             -1          -1
2*p   + ( - p + a)   + (p - 1)

laplace(one(k*t) + sin(a*t) - cos(b*t) - e**t, t, p);


       2    2 -1    -1     2    2 -1            -1
 - p*(p  + b )   + p   + (p  + a )  *a - (p - 1)

laplace(sqrt(x),x,p);


               - 3/2
1/2*sqrt(pi)*p
 laplace(x**(1/2),x,p);


               - 3/2
1/2*sqrt(pi)*p
 on mcd;


laplace(x**(-1/2),x,p);


 sqrt(pi)
----------
 sqrt(p)
 laplace(x**(5/2),x,p);


 15*sqrt(pi)
--------------
            3
 8*sqrt(p)*p

laplace(-1/4*x**2*c*sqrt(x), x, p);


  - 15*sqrt(pi)*c
------------------
              3
  32*sqrt(p)*p


% Elementary functions with argument k*x - tau,
%   where k>0, tau>=0, x is object var.

laplace(cos(x-a),x,p);


       p
---------------
  p*a   2
 e   *(p  + 1)

laplace(one(k*x-tau),x,p);


      1
--------------
  (p*tau)/k
 e         *p

laplace(sinh(k*x-tau),x,p);


           - k
-------------------------
  (p*tau)/k      2    2
 e         *( - p  + k )
 laplace(sinh(k*x),x,p);


     - k
------------
     2    2
  - p  + k

laplace((a*x-b)**c,x,p);


  c
 a *gamma(c + 1)
-----------------
   c  (p*b)/a
  p *e       *p

% But ...
off mcd;

 laplace((a*x-b)**2,x,p);


 -3   2  2                2
p  *(p *b  - 2*p*a*b + 2*a )
 on mcd;


laplace(sin(2*x-3),x,p);


         2
-------------------
  (3*p)/2   2
 e       *(p  + 4)

on lmon;

 laplace(sin(2*x-3),x,p);


         2
-------------------
  (3*p)/2   2
 e       *(p  + 4)
 off lmon;


off mcd;

 laplace(cosh(t-a) - sin(3*t-5), t, p);


  - p*a     2     -1       - 5/3*p   2     -1
e      *p*(p  - 1)   - 3*e        *(p  + 9)
 on mcd;



% More complicated examples - multiplication of functions.
% We use here on lmon - a new switch that forces all
% trigonometrical functions which depend on object var
% to be represented as exponents.

laplace(x*e**(a*x)*cos(k*x), x, p);


                           2            2    2
                          p  - 2*p*a + a  - k
-------------------------------------------------------------------------
  4      3        2  2      2  2        3          2    4      2  2    4
 p  - 4*p *a + 6*p *a  + 2*p *k  - 4*p*a  - 4*p*a*k  + a  + 2*a *k  + k

laplace(x**(1/2)*e**(a*x), x, p);


        - sqrt(pi)
--------------------------
 2*sqrt(p - a)*( - p + a)

laplace(-1/4*e**(a*x)*(x-k)**(-1/2), x, p);


               a*k
   - sqrt(pi)*e
--------------------
    p*k
 4*e   *sqrt(p - a)

laplace(x**(5/2)*e**(a*x), x, p);


                 - 15*sqrt(pi)
----------------------------------------------
                    3      2          2    3
 8*sqrt(p - a)*( - p  + 3*p *a - 3*p*a  + a )

laplace((a*x-b)**c*e**(k*x)*const/2, x, p);


     1   (b*k)/a  c
  - ---*e       *a *gamma(c + 1)*const
     2
---------------------------------------
      (p*b)/a        c
     e       *(p - k) *( - p + k)

off mcd;

 laplace(x*e**(a*x)*sin(7*x)/c*3, x, p);


     2            2      -2  -1
42*(p  - 2*p*a + a  + 49)  *c  *(p - a)
 on mcd;


laplace(x*e**(a*x)*sin(k*x-tau), x, p);


  (a*tau)/k   2                            2                2         (p*tau)/k
(e         *(p *tau - 2*p*a*tau + 2*p*k + a *tau - 2*a*k + k *tau))/(e

      4      3        2  2      2  2        3          2    4      2  2    4
   *(p  - 4*p *a + 6*p *a  + 2*p *k  - 4*p*a  - 4*p*a*k  + a  + 2*a *k  + k ))

% The next is unknown if lmon is off.
laplace(sin(k*x)*cosh(k*x), x, p);


*** Laplace for cosh(x*k)*sin(x*k) not known - try ON LMON 

laplace(cosh(k*x)*sin(k*x),x,p)

laplace(x**(1/2)*sin(k*x), x, p);


*** Laplace for sqrt(x)*sin(x*k) not known - try ON LMON 

laplace(sqrt(x)*sin(k*x),x,p)

on lmon;

  % But now is OK.
laplace(x**(1/2)*sin(a*x)*cos(a*b), x, p);


(sqrt(pi)*cos(a*b)

 *( - sqrt(p - a*i)*p + sqrt(p + a*i)*p + sqrt(p - a*i)*a*i + sqrt(p + a*i)*a*i)

                                      2    2
 )/(4*sqrt(p + a*i)*sqrt(p - a*i)*i*(p  + a ))

laplace(sin(x)*cosh(x), x, p);


  2
 p  + 2
--------
  4
 p  + 4

laplace(sin(k*x)*cosh(k*x), x, p);


     2      2
 k*(p  + 2*k )
---------------
    4      4
   p  + 4*k

% Off exp leads to very messy output in this case.
% off exp; laplace(sin(k*x-t)*cosh(k*x-t), x, p); on exp;
laplace(sin(k*x-t)*cosh(k*x-t), x, p);


        2      2
    k*(p  + 2*k )
----------------------
  (p*t)/k   4      4
 e       *(p  + 4*k )

laplace(cos(x)**2,x,p);


    2
   p  + 2
------------
     2
 p*(p  + 4)
laplace(c*cos(k*x)**2,x,p);


     2      2
 c*(p  + 2*k )
---------------
     2      2
 p*(p  + 4*k )

laplace(c*cos(2/3*x)**2, x, p);


     2    8
 c*(p  + ---)
          9
---------------
     2    16
 p*(p  + ----)
          9

laplace(5*sinh(x)*e**(a*x)*x**3, x, p);


       3      2          2        3         8      7         6  2      6
(120*(p  - 3*p *a + 3*p*a  + p - a  - a))/(p  - 8*p *a + 28*p *a  - 4*p

          5  3       5         4  4       4  2      4       3  5       3  3
    - 56*p *a  + 24*p *a + 70*p *a  - 60*p *a  + 6*p  - 56*p *a  + 80*p *a

          3         2  6       2  4       2  2      2        7         5
    - 24*p *a + 28*p *a  - 60*p *a  + 36*p *a  - 4*p  - 8*p*a  + 24*p*a

            3            8      6      4      2
    - 24*p*a  + 8*p*a + a  - 4*a  + 6*a  - 4*a  + 1)

off exp;

 laplace(sin(2*x-3)*cosh(7*x-5), x, p);


                2  11    2         11              11
               p *e   + p  + 14*p*e   - 14*p + 53*e   + 53
-------------------------------------------------------------------------
  (3*p + 1)/2                                                          5
 e           *(p + 7 + 2*i)*(p + 7 - 2*i)*(p - 7 + 2*i)*(p - 7 - 2*i)*e
 on exp;


laplace(sin(a*x-b)*cosh(c*x-d), x, p);


*** Laplace for  - 1/4*one((x*a - b)/a)*one((x*c - d)/c)*i**(-1) not known 

*** Laplace for 1/4*one((x*a - b)/a)*one((x*c - d)/c)*i**(-1) not known 

             b*i      a*x - b        c*x - d     2*c*x    2*d
          - e   *one(---------)*one(---------)*(e      + e   )
                         a              c
laplace(-------------------------------------------------------,x,p)
                            a*i*x + c*x + d
                         4*e               *i

             a*i*x      a*x - b        c*x - d     2*c*x    2*d
            e     *one(---------)*one(---------)*(e      + e   )
                           a              c
 + laplace(------------------------------------------------------,x,p)
                                b*i + c*x + d
                             4*e             *i

% To solve this problem we must tell the program which one-function
% is rightmost shifted.  However, in REDUCE 3.4, this rule is still
% not sufficient.
for all x let one(x-b/a)*one(x-d/c) = one(x-b/a);


laplace(sin(a*x-b)*cosh(c*x-d), x, p);


     (2*b*c)/a  2    2*d  2      (2*b*c)/a          2*d        (2*b*c)/a  2
(a*(e         *p  + e   *p  + 2*e         *p*c - 2*e   *p*c + e         *a

        (2*b*c)/a  2    2*d  2    2*d  2       (p*b + a*d + b*c)/a
     + e         *c  + e   *a  + e   *c ))/(2*e

      4      2  2      2  2    4      2  2    4
   *(p  + 2*p *a  - 2*p *c  + a  + 2*a *c  + c ))

for all x clear one(x-b/a)*one(x-d/c) ;


off lmon;



% Floating point arithmetic.
% laplace(3.5/c*sin(2.3*x-4.11)*e**(1.5*x), x, p);
on rounded;


laplace(3.5/c*sin(2.3*x-4.11)*e**(1.5*x), x, p);


                   117.461059957
----------------------------------------------------
              1.78695652174*p     2
 2.71828182846               *c*(p  - 3.0*p + 7.54)

laplace(x**2.156,x,p);


 gamma(3.156)
--------------
     3.156
    p

laplace(x**(-0.5),x,p);


 gamma(0.5)
------------
     0.5
    p

off rounded;

 laplace(x**(-0.5),x,p);


 sqrt(pi)
----------
 sqrt(p)
 on rounded;


laplace(x*e**(2.35*x)*cos(7.42*x), x, p);


                   2
                  p  - 4.7*p - 49.5339
---------------------------------------------------------
  4        3             2
 p  - 9.4*p  + 143.2478*p  - 569.44166*p + 3669.80312521

laplace(x*e**(2.35*x)*cos(7.42*x-74.2), x, p);


                 3                      2
(160664647206.0*p  - 1.11661929808e+12*p  + 1.14319162408e+13*p

                                     10.0*p
  - 2.36681205089e+13)/(2.71828182846

      4        3             2
   *(p  - 9.4*p  + 143.2478*p  - 569.44166*p + 3669.80312521))

% Higher precision works, but uses more memory.
% precision 20; laplace(x**2.156,x,p);
% laplace(x*e**(2.35*x)*cos(7.42*x-74.2), x, p);
off rounded;



% Integral from 0 to x, where x is object var.
% Syntax is intl(<expr>,<var>,0,<obj.var>).

laplace(c1/c2*intl(2*y**2,y,0,x), x,p);


 4*c1
-------
  4
 p *c2

off mcd;

 laplace(intl(e**(2*y)*y**2+sqrt(y),y,0,x),x,p);


 -1           -3                  - 3/2
p  *(2*(p - 2)   + 1/2*sqrt(pi)*p      )
 on mcd;


laplace(-2/3*intl(1/2*y*e**(a*y)*sin(k*y),y,0,x), x, p);


                                   2       2
                            k*( - ---*p + ---*a)
                                   3       3
-----------------------------------------------------------------------------
     4      3        2  2      2  2        3          2    4      2  2    4
 p*(p  - 4*p *a + 6*p *a  + 2*p *k  - 4*p*a  - 4*p*a*k  + a  + 2*a *k  + k )


% Use of delta function and derivatives.

laplace(-1/2*delta(x), x, p);


    1
 - ---
    2
 laplace(delta(x-tau), x, p);


   1
--------
  p*tau
 e

laplace(c*cos(k*x)*delta(x),x,p);


c

laplace(e**(a*x)*delta(x), x, p);


1

laplace(c*x**2*delta(x), x, p);


0

laplace(-1/4*x**2*delta(x-pi), x, p);


     1    2
  - ---*pi
     4
------------
    p*pi
   e

laplace(cos(2*x-3)*delta(x-pi),x,p);


 cos(3)
--------
  p*pi
 e

laplace(e**(-b*x)*delta(x-tau), x, p);


      1
--------------
  tau*(p + b)
 e

on lmon;


laplace(cos(2*x)*delta(x),x,p);


1

laplace(c*x**2*delta(x), x, p);


0

laplace(c*x**2*delta(x-pi), x, p);


     2
 c*pi
-------
  p*pi
 e

laplace(cos(a*x-b)*delta(x-pi),x,p);


 cos(a*pi - b)
---------------
      p*pi
     e

laplace(e**(-b*x)*delta(x-tau), x, p);


      1
--------------
  tau*(p + b)
 e

off lmon;



laplace(2/3*df(delta x,x),x,p);


 2
---*p
 3

off exp;

 laplace(e**(a*x)*df(delta x,x,5), x, p);


             5
 - ( - p + a)
 on exp;


laplace(df(delta(x-a),x), x, p);


  p
------
  p*a
 e

laplace(e**(k*x)*df(delta(x),x), x, p);


p - k

laplace(e**(k*x)*c*df(delta(x-tau),x,2), x, p);


  k*tau     2            2
 e     *c*(p  - 2*p*k + k )
----------------------------
            p*tau
           e

on lmon;

laplace(e**(k*x)*sin(a*x)*df(delta(x-t),x,2),x,p);


  k*t   1   2*a*i*t  2    1   2    2*a*i*t          2*a*i*t
(e   *(---*e       *p  - ---*p  - e       *p*a*i - e       *p*k - p*a*i + p*k
        2                 2

           1   2*a*i*t  2    2*a*i*t          1   2*a*i*t  2    1   2
        - ---*e       *a  + e       *a*i*k + ---*e       *k  + ---*a  + a*i*k
           2                                  2                 2

           1   2     t*(p + a*i)
        - ---*k ))/(e           *i)
           2
off lmon;



% But if tau is positive, Laplace transform is not defined.

laplace(e**(a*x)*delta(x+tau), x, p);


*** Laplace for delta(x + tau) not known - try ON LMON 

         a*x
laplace(e   *delta(tau + x),x,p)

laplace(2*c*df(delta(x+tau),x), x, p);


*** Laplace for df(delta(x + tau),x) not known - try ON LMON 

laplace(2*df(delta(tau + x),x)*c,x,p)

laplace(e**(k*x)*df(delta(x+tau),x,3), x, p);


*** Laplace for df(delta(x + tau),x,3) not known - try ON LMON 

         k*x
laplace(e   *df(delta(tau + x),x,3),x,p)


% Adding new let rules for Laplace operator. Note the syntax.

for all x let laplace(log(x),x) = -log(gam*il!&)/il!&;


laplace(-log(x)*a/4, x, p);


  1
 ---*log(p*gam)*a
  4
------------------
        p
 laplace(-log(x),x,p);


 log(p*gam)
------------
     p

laplace(a*log(x)*e**(k*x), x, p);


 log(gam*(p - k))*a
--------------------
       - p + k

for all x clear laplace(log(x),x);



operator f;

 for all x let
    laplace(df(f(x),x),x) = il!&*laplace(f(x),x) - sub(x=0,f(x));


for all x,n such that numberp n and fixp n let
    laplace(df(f(x),x,n),x) = il!&**n*laplace(f(x),x) -
      for i:=n-1 step -1 until 0 sum
        sub(x=0, df(f(x),x,n-1-i)) * il!&**i ;


for all x let laplace(f(x),x) = f(il!&);



laplace(1/2*a*df(-2/3*f(x)*c,x), x,p);


         1            1
a*c*( - ---*p*f(p) + ---*f(0))
         3            3

laplace(1/2*a*df(-2/3*f(x)*c,x,4), x,p);


         1   4         1   3         1   2
a*c*( - ---*p *f(p) + ---*p *f(0) + ---*p *sub(x=0,df(f(x),x))
         3             3             3

         1                             1
      + ---*p*sub(x=0,df(f(x),x,2)) + ---*sub(x=0,df(f(x),x,3)))
         3                             3

laplace(1/2*a*e**(k*x)*df(-2/3*f(x)*c,x,2), x,p);


         1   2             2                  1            1            2
a*c*( - ---*p *f(p - k) + ---*p*f(p - k)*k + ---*p*f(0) - ---*f(p - k)*k
         3                 3                  3            3

         1            1
      - ---*f(0)*k + ---*sub(x=0,df(f(x),x)))
         3            3

clear f;



% Or if the boundary conditions are known and assume that
% f(i,0)=sub(x=0,df(f(x),x,i)) the above may be overwritten as:
operator f;

 for all x let
    laplace(df(f(x),x),x) = il!&*laplace(f(x),x) - f(0,0);


for all x,n such that numberp n and fixp n let
    laplace(df(f(x),x,n),x) = il!&**n*laplace(f(x),x) -
      for i:=n-1 step -1 until 0 sum il!&**i * f(n-1-i,0);


for all x let laplace(f(x),x) = f(il!&);


let f(0,0)=0, f(1,0)=1, f(2,0)=2, f(3,0)=3;


laplace(1/2*a*df(-2/3*f(x)*c,x), x,p);


    1
 - ---*p*f(p)*a*c
    3

laplace(1/2*a*df(-2/3*f(x)*c,x,4), x,p);


         1   4         1   2    2
a*c*( - ---*p *f(p) + ---*p  + ---*p + 1)
         3             3        3

clear f(0,0), f(1,0), f(2,0), f(3,0);

 clear f;



% Very complicated examples.

on lmon;


laplace(sin(a*x-b)**2, x, p);


     (p*b)/a  2    (p*b)/a  2      (p*b)/a  2
  - e       *p  + e       *p  + 4*e       *a
----------------------------------------------
             (2*p*b)/a     2      2
          2*e         *p*(p  + 4*a )

off mcd;

 laplace(x**3*(sin x)**4*e**(5*k*x)*c/2, x,p);


                          -4                       -4                         -4
c*(3/16*( - p + 4*i + 5*k)   + 3/16*(p + 4*i - 5*k)   - 3/4*( - p + 2*i + 5*k)

                         -4                   -4
    - 3/4*(p + 2*i - 5*k)   + 9/8*( - p + 5*k)  )

a:=(sin x)**4*e**(5*k*x)*c/2;


          5*k*x       4
a := 1/2*e     *sin(x) *c
 laplace(x**3*a,x,p);


                          -4                       -4                         -4
c*(3/16*( - p + 4*i + 5*k)   + 3/16*(p + 4*i - 5*k)   - 3/4*( - p + 2*i + 5*k)

                         -4                   -4
    - 3/4*(p + 2*i - 5*k)   + 9/8*( - p + 5*k)  )
 clear a;

 on mcd;


% And so on, but is very time consuming.
% laplace(e**(k*x)*x**2*sin(a*x-b)**2, x, p);
% for all x let one(a*x-b)*one(c*x-d) = one(c*x-d);
% laplace(x*e**(-2*x)*cos(a*x-b)*sinh(c*x-d), x, p);
% for all x clear one(a*x-b)*one(c*x-d) ;
% laplace(x*e**(c*x)*sin(k*x)**3*cosh(x)**2*cos(a*x), x, p);
off lmon;



% Error messages.

laplace(sin(-x),x,p);


***** Laplace induces one( - x)  which is not allowed 

laplace( - sin(x),x,p)

on lmon;

 laplace(sin(-a*x), x, p);


***** Laplace induces one( - x*a)  which is not allowed 

laplace( - sin(a*x),x,p)
 off lmon;


laplace(e**(k*x**2), x, p);


*** Laplace for e**(x**2*k) not known - try ON LMON 

            2
         k*x
laplace(e    ,x,p)

laplace(sin(-a*x+b)*cos(c*x+d), x, p);


*** Laplace for  - cos(x*c + d)*sin(x*a - b) not known - try ON LMON 

laplace( - cos(c*x + d)*sin(a*x - b),x,p)

laplace(x**(-5/2),x,p);


*** Laplace for x**( - 5/2) not known - try ON LMON 

             1
laplace(------------,x,p)
                  2
         sqrt(x)*x

% With int arg, can't be shifted.
laplace(intl(y*e**(a*y)*sin(k*y-tau),y,0,x), x, p);


*** Laplace for sin(x*k - tau) not allowed 

          a*x
 laplace(e   *sin(k*x - tau)*x,x,p)
------------------------------------
                 p

laplace(cosh(x**2), x, p);


*** Laplace for cosh(x**2) not known - try ON LMON 

              2
laplace(cosh(x ),x,p)

laplace(3*x/(x**2-5*x+6),x,p);


*** Laplace for (x**2 - 5*x + 6)**(-1) not known - try ON LMON 

             3*x
laplace(--------------,x,p)
          2
         x  - 5*x + 6

laplace(1/sin(x),x,p);


*** Laplace for sin(x)**(-1) not known - try ON LMON 

           1
laplace(--------,x,p)
         sin(x)
   % But ...
laplace(x/sin(-3*a**2),x,p);


      - 1
--------------
  2        2
 p *sin(3*a )

% Severe errors.
% laplace(sin x,x,cos y);
% laplace(sin x,x,y+1);
% laplace(sin(x+1),x+1,p);


Comment  Examples of Inverse Laplace transformations;


symbolic(ordl!* := nil);

   % To nullify previous order declarations.

order t;



% Elementary ratio of polynomials.

invlap(1/p, p, t);


1

invlap(1/p**3, p, t);


  2
 t
----
 2

invlap(1/(p-a), p, t);


 t*a
e
 invlap(1/(2*p-a),p,t);


  (t*a)/2
 e
----------
    2
 invlap(1/(p/2-a),p,t);


   2*t*a
2*e

invlap(e**(-k*p)/(p-a), p, t);


  t*a
 e
------
  a*k
 e
 invlap(b**(-k*p)/(p-a), p, t);


  t*a
 e
------
  a*k
 b

invlap(1/(p-a)**3, p, t);


  t*a  2
 e   *t
---------
    2

invlap(1/(c*p-a)**3, p, t);


  (t*a)/c  2
 e       *t
-------------
       3
    2*c
 invlap(1/(p/c-a)**3, p, t);


  t*a*c  2  3
 e     *t *c
--------------
      2

invlap((c*p-a)**(-1)/(c*p-a)**2, p, t);


  (t*a)/c  2
 e       *t
-------------
       3
    2*c

invlap(c/((p/c-a)**2*(p-a*c)), p, t);


  t*a*c  2  3
 e     *t *c
--------------
      2

invlap(1/(p*(p-a)), p, t);


  t*a
 e    - 1
----------
    a

invlap(c/((p-a)*(p-b)), p, t);


     t*a    t*b
 c*(e    - e   )
-----------------
      a - b

invlap(p/((p-a)*(p-b)), p, t);


  t*a      t*b
 e   *a - e   *b
-----------------
      a - b

off mcd;

 invlap((p+d)/(p*(p-a)), p, t);


 t*a  -1      t*a    -1
e   *a  *d + e    - a  *d

invlap((p+d)/((p-a)*(p-b)), p, t);


       -1   t*a      t*a      t*b      t*b
(a - b)  *(e   *a + e   *d - e   *b - e   *d)

invlap(1/(e**(k*p)*p*(p+1)), p, t);


     - t + k
 - e         + one(t - k)
 on mcd;


off exp;

 invlap(c/(p*(p+a)**2), p, t);


                t*a
  - (a*t + 1 - e   )*c
-----------------------
         t*a  2
        e   *a
 on exp;


invlap(1, p, t);


delta(t)
 invlap(c1*p/c2, p, t);


 df(delta(t),t)*c1
-------------------
        c2

invlap(p/(p-a), p, t);


            t*a
delta(t) + e   *a
 invlap(c*p**2, p, t);


df(delta(t),t,2)*c

invlap(p**2*e**(-a*p)*c, p, t);


sub(t=t - a,df(delta(t),t,2))*c

off mcd;

invlap(e**(-a*p)*(1/p**2-p/(p-1))+c/p, p, t);


                    t - a
t - delta(t - a) - e      - a + c
on mcd;


invlap(a*p**2-2*p+1, p, x);


delta(x) + df(delta(x),x,2)*a - 2*df(delta(x),x)


% P to non-integer power in denominator - i.e. gamma-function case.

invlap(1/sqrt(p), p, t);


        1
------------------
 sqrt(t)*sqrt(pi)
 invlap(1/sqrt(p-a), p, t);


        t*a
       e
------------------
 sqrt(t)*sqrt(pi)

invlap(c/(p*sqrt(p)), p, t);


 2*sqrt(t)*c
-------------
  sqrt(pi)
 invlap(c*sqrt(p)/p**2, p, t);


 2*sqrt(t)*c
-------------
  sqrt(pi)

invlap((p-a)**(-3/2), p, t);


            t*a
 2*sqrt(t)*e
----------------
    sqrt(pi)

invlap(sqrt(p-a)*c/(p-a)**2, p, t);


            t*a
 2*sqrt(t)*e   *c
------------------
     sqrt(pi)

invlap(1/((p-a)*b*sqrt(p-a)), p, t);


            t*a
 2*sqrt(t)*e
----------------
   sqrt(pi)*b

invlap((p/(c1-3)-a)**(-3/2), p, t);


            t*a*c1
 2*sqrt(t)*e      *sqrt(c1 - 3)*(c1 - 3)
-----------------------------------------
                       3*t*a
             sqrt(pi)*e

invlap(1/((p/(c1-3)-a)*b*sqrt(p/(c1-3)-a)), p, t);


            t*a*c1
 2*sqrt(t)*e      *sqrt(c1 - 3)*(c1 - 3)
-----------------------------------------
                      3*t*a
            sqrt(pi)*e     *b

invlap((p*2-a)**(-3/2), p, t);


          (t*a)/2
 sqrt(t)*e
------------------
 sqrt(pi)*sqrt(2)

invlap(sqrt(2*p-a)*c/(p*2-a)**2, p, t);


          (t*a)/2
 sqrt(t)*e       *sqrt(2)*c
----------------------------
         2*sqrt(pi)

invlap(c/p**(7/2), p, t);


            2
 8*sqrt(t)*t *c
----------------
  15*sqrt(pi)
 invlap(p**(-7/3), p, t);


    1/3
   t   *t
------------
        7
 gamma(---)
        3

invlap(gamma(b)/p**b,p,t);


  b
 t
----
 t
 invlap(c*gamma(b)*(p-a)**(-b),p,t);


  b  t*a
 t *e   *c
-----------
     t

invlap(e**(-k*p)/sqrt(p-a), p, t);


            t*a
           e
---------------------------
           a*k
 sqrt(pi)*e   *sqrt(t - k)


% Images that give elementary object functions.
% Use of new switches lmon, lhyp.

invlap(k/(p**2+k**2), p, t);


  2*t*i*k
 e        - 1
--------------
     t*i*k
  2*e     *i

% This is made more readable by :
on ltrig;

 invlap(k/(p**2+k**2), p, t);


sin(t*k)

invlap(p/(p**2+1), p, t);


cos(t)

invlap((p**2-a**2)/(p**2+a**2)**2, p, t);


t*cos(t*a)

invlap(p/(p**2+a**2)**2, p, t);


 t*sin(t*a)
------------
    2*a

invlap((p-a)/((p-a)**2+b**2), p, t);


 t*a
e   *cos(t*b)
 off ltrig;


on lhyp;

 invlap(s/(s**2-k**2), s, t);


cosh(t*k)

invlap(e**(-tau/k*p)*p/(p**2-k**2), p, t);


cosh(t*k - tau)
 off lhyp;


% But it is not always possible to convert expt. functions, e.g.:
on lhyp;

 invlap(k/((p-a)**2-k**2), p, t);


sinh(t*k)*(cosh(t*a) + sinh(t*a))
 off lhyp;


on ltrig;

 invlap(e**(-tau/k*p)*k/(p**2+k**2), p, t);


  2*t*i*k    2*i*tau
 e        - e
---------------------
    i*(t*k + tau)
 2*e             *i
 off ltrig;


% In such situations use the default switches:
invlap(k/((p-a)**2-k**2), p, t);


  t*a   2*t*k
 e   *(e      - 1)
-------------------
         t*k
      2*e
 % i.e. e**(a*t)*cosh(k*t).
invlap(e**(-tau/k*p)*k/(p**2+k**2), p, t);


  2*t*i*k    2*i*tau
 e        - e
---------------------
    i*(t*k + tau)
 2*e             *i
 % i.e. sin(k*t-tau).

% More complicated examples.

off exp,mcd;

 invlap((p+d)/(p**2*(p-a)), p, t);


                       t*a           -2
 - ((d*t + 1)*a + d - e   *(a + d))*a

invlap(e**(-tau/k*p)*c/(p*(p-a)**2), p, t);


          -1
      - (k  *tau - t)*a    -1                            -1        -2
 - (e                  *((k  *tau - t)*a + 1) - one(t - k  *tau))*a  *c

invlap(1/((p-a)*(p-b)*(p-c)), p, t);


     t*b               2       -1    t*c                     2 -1
 - (e   *(a*b - a*c - b  + b*c)   - e   *(a*b - a*c - b*c + c )

        t*a   2                   -1
     - e   *(a  - a*b - a*c + b*c)  )

invlap((p**2+g*p+d)/(p*(p-a)**2), p, t);


     t*a   -2           -2      t*a       -1
 - (e   *(a  *d - 1) - a  *d - e   *(a + a  *d + g)*t)
 on exp,mcd;


invlap(k*c**(-b*p)/((p-a)**2+k**2), p, t);


  t*a      2*b*i*k    2*t*i*k
 e   *( - c        + e       )
-------------------------------
       t*i*k  a*b + b*i*k
    2*e     *c           *i

on ltrig;

 invlap(c/(p**2*(p**2+a**2)), p, t);


 c*(t*a - sin(t*a))
--------------------
          3
         a

invlap(1/(p**2-p+1), p, t);


    t/2      sqrt(3)*t
 2*e   *sin(-----------)
                 2
-------------------------
         sqrt(3)
 invlap(1/(p**2-p+1)**2, p, t);


    t/2              sqrt(3)*t                    sqrt(3)*t
 2*e   *( - 3*t*cos(-----------) + 2*sqrt(3)*sin(-----------))
                         2                            2
---------------------------------------------------------------
                               9

invlap(2*a**2/(p*(p**2+4*a**2)), p, t);


  - cos(2*t*a) + 1
-------------------
         2

% This is (sin(a*t))**2 and you can get this by using the let rules :
for all x let sin(2*x)=2*sin x*cos x, cos(2*x)=(cos x)**2-(sin x)**2,
(cos x)**2 =1-(sin x)**2;


invlap(2*a**2/(p*(p**2+4*a**2)), p, t);


        2
sin(t*a)

for all x clear sin(2*x),cos(2*x),cos(x)**2;

  off ltrig;


on lhyp;

invlap((p**2-2*a**2)/(p*(p**2-4*a**2)),p,t);


 cosh(2*t*a) + 1
-----------------
        2

off lhyp;

 % Analogously, the above is (cosh(a*t))**2.

% Floating arithmetic.

invlap(2.55/((0.5*p-2.0)*(p-3.3333)), p, t);


            (33333*t)/10000    4*t
 51000*( - e                + e   )
------------------------------------
                6667

on rounded;


invlap(2.55/((0.5*p-2.0)*(p-3.3333)), p, t);


                           4.0*t                              3.3333*t
7.64961751912*2.71828182846      - 7.64961751912*2.71828182846

invlap(1.5/sqrt(p-0.5), p, t);


                  0.5*t
 1.5*2.71828182846
------------------------
     0.5
    t   *gamma(0.5)

invlap(2.75*p**2-0.5*p+e**(-0.9*p)/p, p, t);


2.75*df(delta(t),t,2) - 0.5*df(delta(t),t) + one(t - 0.9)

invlap(1/(2.0*p-3.0)**3, p, t);


                    1.5*t  2
0.0625*2.71828182846     *t
 invlap(1/(2.0*p-3.0)**(3/2), p, t);


                 0.5              1.5*t
 0.353553390593*t   *2.71828182846
----------------------------------------
               gamma(1.5)

invlap(1/(p**2-5.0*p+6), p, t);


             3.0*t                2.0*t
2.71828182846      - 2.71828182846

off rounded;



% Adding new let rules for the invlap operator. note the syntax:

for all x let invlap(log(gam*x)/x,x) = -log(lp!&);


invlap(-1/2*log(gam*p)/p, p, t);


 log(t)
--------
   2

invlap(-e**(-a*p)*log(gam*p)/(c*p), p, t);


 log(t - a)
------------
     c

for all x clear invlap(1/x*log(gam*x),x);



% Very complicated examples and use of factorizer.

off exp,mcd;

 invlap(c**(-k*p)*(p**2+g*p+d)/(p**2*(p-a)**3), p, t);


   - (log(c)*k - t)*a                   -4
(e                    - 1)*(a*g + 3*d)*a

         - (log(c)*k - t)*a                  2   -1      -2
 + 1/2*e                   *( - t + log(c)*k) *(a  *g + a  *d + 1)

      - (log(c)*k - t)*a                                  -3
 + (e                   *(a*g + 2*d) + d)*(log(c)*k - t)*a

on exp,mcd;


invlap(1/(2*p**3-5*p**2+4*p-1), p, t);


 t        t/2      t
e *t + 2*e    - 2*e

on ltrig,lhyp;

 invlap(1/(p**4-a**4), p, t);


  - sin(t*a) + sinh(t*a)
-------------------------
             3
          2*a

invlap(1/((b-3)*p**4-a**4*(2+b-5)), p, t);


  - sin(t*a) + sinh(t*a)
-------------------------
         3
      2*a *(b - 3)
 off ltrig,lhyp;


% The next three examples are the same:
invlap(c/(p**3/8-9*p**2/4+27/2*p-27)**2,p,t);


      6*t  5
 243*e   *t *c
---------------
      40
invlap(c/(p/2-3)**6,p,t);


    6*t  5
 8*e   *t *c
-------------
     15

off exp;

 a:=(p/2-3)**6;


             6
      (p - 6)
a := ----------
         64
 on exp;

 invlap(c/a, p, t);


    6*t  5
 8*e   *t *c
-------------
     15
 clear a;


% The following two examples are the same :
invlap(c/(p**4+2*p**2+1)**2, p, t);


     2*t*i  3    3      2*t*i  2        2         2*t*i                2*t*i
(c*(e     *t  + t  + 6*e     *t *i - 6*t *i - 15*e     *t - 15*t - 15*e     *i

                   t*i
     + 15*i))/(96*e   )
 invlap(c/((p-i)**4*(p+i)**4),p,t);


     2*t*i  3    3      2*t*i  2        2         2*t*i                2*t*i
(c*(e     *t  + t  + 6*e     *t *i - 6*t *i - 15*e     *t - 15*t - 15*e     *i

                   t*i
     + 15*i))/(96*e   )

% The following three examples are the same :
invlap(e**(-k*p)/(2*p-3)**6, p, t);


  (3*t)/2   5      4         3  2       2  3        4    5
 e       *(t  - 5*t *k + 10*t *k  - 10*t *k  + 5*t*k  - k )
------------------------------------------------------------
                             (3*k)/2
                       7680*e

invlap(e**(-k*p)/(4*p**2-12*p+9)**3, p, t);


  (3*t)/2   5      4         3  2       2  3        4    5
 e       *(t  - 5*t *k + 10*t *k  - 10*t *k  + 5*t*k  - k )
------------------------------------------------------------
                             (3*k)/2
                       7680*e

invlap(e**(-k*p)/(8*p**3-36*p**2+54*p-27)**2, p, t);


  (3*t)/2   5      4         3  2       2  3        4    5
 e       *(t  - 5*t *k + 10*t *k  - 10*t *k  + 5*t*k  - k )
------------------------------------------------------------
                             (3*k)/2
                       7680*e


% Error messages.

invlap(e**(a*p)/p, p, t);


*** Invlap for e**(p*a)/p not known 

         a*p
        e
invlap(------,p,t)
         p

invlap(c*p*sqrt(p), p, t);


*** Invlap for sqrt(p)*p not known 

invlap(sqrt(p)*c*p,p,t)

invlap(sin(p), p, t);


*** Invlap for sin(p) not known 

invlap(sin(p),p,t)

invlap(1/(a*p**3+b*p**2+c*p+d),p,t);


*** Invlap for (p**3*a + p**2*b + p*c + d)**(-1) not known 

                  1
invlap(-----------------------,p,t)
           3      2
        a*p  + b*p  + c*p + d

invlap(1/(p**2-p*sin(p)+a**2),p,t);


*** Invlap for (p**2 - p*sin(p) + a**2)**(-1) not known 

                - 1
invlap(--------------------,p,t)
                    2    2
        sin(p)*p - a  - p

on rounded;

 invlap(1/(p**3-1), p, t);


*** Invlap for (p**3 - 1)**(-1) not known 

          1
invlap(--------,p,t)
         3
        p  - 1
 off rounded;


% Severe errors:
%invlap(1/(p**2+1), p+1, sin(t) );
%invlap(p/(p+1)**2, sin(p), t);

end;
(TIME:  laplace 8570 8939)