Tue Feb 10 12:28:22 2004 run on Linux
% 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);
a
-----------
2 2
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);
(b*k)/a c
- e *a *gamma(c + 1)*const
-----------------------------------
(p*b)/a c
2*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*i - sqrt(p + a*i)*p*i + sqrt(p - a*i)*a + sqrt(p + a*i)*a))/(
2 2
4*sqrt(p + a*i)*sqrt(p - a*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
c*(9*p + 8)
---------------
2
p*(9*p + 16)
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
a*i*x a*x - b c*x - d 2*c*x 2*d
- e *one(---------)*one(---------)*i*(e + e )
a c
laplace(-----------------------------------------------------------,x,p)
b*i + c*x + d
4*e
b*i a*x - b c*x - d 2*c*x 2*d
e *one(---------)*one(---------)*i*(e + e )
a c
+ laplace(------------------------------------------------------,x,p)
a*i*x + c*x + d
4*e
% 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);
2.32056900246
---------------
3.156
p
laplace(x**(-0.5),x,p);
1.77245385091
---------------
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*k*( - p + a)
-------------------------------------------------------------------------------
4 3 2 2 2 2 3 2 4 2 2 4
3*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);
2
- pi
---------
p*pi
4*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 2*a*i*t 2 2 2*a*i*t 2*a*i*t
(e *( - e *p *i + p *i - 2*e *p*a + 2*e *p*i*k - 2*p*a
2*a*i*t 2 2*a*i*t 2*a*i*t 2 2
- 2*p*i*k + e *a *i + 2*e *a*k - e *i*k - a *i
2 t*(p + a*i)
+ 2*a*k + i*k ))/(2*e )
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);
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);
a*c*( - p*f(p) + f(0))
------------------------
3
laplace(1/2*a*df(-2/3*f(x)*c,x,4), x,p);
4 3 2
(a*c*( - p *f(p) + p *f(0) + p *sub(x=0,df(f(x),x)) + p*sub(x=0,df(f(x),x,2))
+ sub(x=0,df(f(x),x,3))))/3
laplace(1/2*a*e**(k*x)*df(-2/3*f(x)*c,x,2), x,p);
2 2
(a*c*( - p *f(p - k) + 2*p*f(p - k)*k + p*f(0) - f(p - k)*k - f(0)*k
+ sub(x=0,df(f(x),x))))/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);
- p*f(p)*a*c
---------------
3
laplace(1/2*a*df(-2/3*f(x)*c,x,4), x,p);
4 2
a*c*( - p *f(p) + p + 2*p + 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);
2
2*a
------------------------
(p*b)/a 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
i*( - e + 1)
---------------------
t*i*k
2*e
% 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
i*( - e + e )
----------------------------
i*(t*k + tau)
2*e
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
i*( - e + e )
----------------------------
i*(t*k + tau)
2*e
% 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 *i*(c - e )
------------------------------
t*i*k a*b + b*i*k
2*e *c
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
0.846284375322*2.71828182846
-----------------------------------
0.5
t
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
0.398942280401*t *2.71828182846
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
8*e *t *c
-------------
15
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 for test: 680 ms, plus GC time: 90 ms