REDUCE 3.4, 15-Jul-91 ...
1:
(LAPLACE)
% 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 + (A - P) + (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*(B + P ) + P + (A + P ) *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)
8
---------------
3
SQRT(P)*P
laplace(-1/4*x**2*c*sqrt(x), x, p);
15
- ----*SQRT(PI)*C
32
--------------------
3
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 4 3 2 2 2 2 3
(P - 2*P*A + A - K )/(P - 4*P *A + 6*P *A + 2*P *K - 4*P*A
2 4 2 2 4
- 4*P*A*K + A + 2*A *K + K )
laplace(x**(1/2)*e**(a*x), x, p);
1
- ---*SQRT(PI)
2
---------------------------
SQRT( - A + P)*( - P + A)
laplace(-1/4*e**(a*x)*(x-k)**(-1/2), x, p);
1 A*K
- ---*SQRT(PI)*E
4
----------------------
P*K
E *SQRT( - A + P)
laplace(x**(5/2)*e**(a*x), x, p);
15
- ----*SQRT(PI)
8
-----------------------------------------------
3 2 2 3
SQRT( - A + P)*( - 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 *( - K + P) *( - P + K)
off mcd;
laplace(x*e**(a*x)*sin(7*x)/c*3, x, p);
2 2 -2 -1
42*(A - 2*A*P + P + 49) *C *(P - A)
on mcd;
laplace(x*e**(a*x)*sin(k*x-tau), x, p);
(A*TAU)/K 2 2 2
(E *(P *TAU - 2*P*A*TAU + 2*P*K + A *TAU - 2*A*K + K *TAU))/(
(P*TAU)/K 4 3 2 2 2 2 3 2
E *(P - 4*P *A + 6*P *A + 2*P *K - 4*P*A - 4*P*A*K
4 2 2 4
+ A + 2*A *K + K ))
% The next is unknown if lmon is off.
laplace(sin(k*x)*cosh(k*x), x, p);
*** Laplace for SIN(X*K)*COSH(X*K) not known - try ON LMON
LAPLACE(SIN(K*X)*COSH(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( - A*I + P)*P + SQRT(A*I + P)*P
+ SQRT( - A*I + P)*A*I + SQRT(A*I + P)*A*I))/(4*SQRT(A*I + P)
2 2
*SQRT( - A*I + P)*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;
laplace(sin(k*x-t)*cosh(k*x-t), x, p);
2*I*T ((P + I*K + K)*T)/K
( - (E *(E *(I*K + K + P)
((P + I*K - K)*T + 2*K*T)/K
+ E *(I*K - K + P))
*(I*K + K - P)*(I*K - K - P) +
((P + I*K + K)*T + (P + I*K - K)*T)/K
E *(
( - (P - I*K + K)*T + 2*K*T)/K
E *(I*K + K - P)
- ((P - I*K - K)*T)/K
+ E *(I*K - K - P))*(I*K + K + P)
*(I*K - K + P)))/(4
((I + 1)*K*T + (P + I*K + K)*T + (P + I*K - K)*T)/K
*E
*(I*K + K + P)*(I*K + K - P)*(I*K - K + P)*(I*K - K - P)*I)
on exp;
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
(120*(P - 3*P *A + 3*P*A + P - A - A))/(P - 8*P *A + 28*P *A
6 5 3 5 4 4 4 2 4
- 4*P - 56*P *A + 24*P *A + 70*P *A - 60*P *A + 6*P
3 5 3 3 3 2 6 2 4 2 2
- 56*P *A + 80*P *A - 24*P *A + 28*P *A - 60*P *A + 36*P *A
2 7 5 3 8 6 4
- 4*P - 8*P*A + 24*P*A - 24*P*A + 8*P*A + A - 4*A + 6*A
2
- 4*A + 1)
off exp;
laplace(sin(2*x-3)*cosh(7*x-5), x, p);
2 11 2 11 11 (3*P + 1)/2
(P *E + P + 14*P*E - 14*P + 53*E + 53)/(E
5
*(2*I + P + 7)*(2*I + P - 7)*(2*I - P + 7)*(2*I - P - 7)*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 + 2*C*X A*X - B C*X - D
LAPLACE(( - E *ONE(---------)*ONE(---------)
A C
B*I + 2*D A*X - B C*X - D
- E *ONE(---------)*ONE(---------))/(4
A C
A*I*X + C*X + D
*E *I),X,P) + LAPLACE((
A*I*X + 2*C*X A*X - B C*X - D
E *ONE(---------)*ONE(---------)
A C
A*I*X + 2*D A*X - B C*X - D
+ E *ONE(---------)*ONE(---------))/(4
A C
B*I + C*X + D
*E *I),X,P)
% 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);
*** Laplace for - 1/4*ONE((X*A - B)/A)*ONE( - A**(-1)*B + X)**(-1)*
ONE(X - A**(-1)*B)*I**(-1) not known
*** Laplace for 1/4*ONE((X*A - B)/A)*ONE( - A**(-1)*B + X)**(-1)*ONE(
X - A**(-1)*B)*I**(-1) not known
B*I + 2*C*X A*X - B B*I + 2*D A*X - B
- E *ONE(---------) - E *ONE(---------)
A A
LAPLACE(------------------------------------------------------------,
A*I*X + C*X + D
4*E *I
X,P) + LAPLACE(
A*I*X + 2*C*X A*X - B A*I*X + 2*D A*X - B
E *ONE(---------) + E *ONE(---------)
A A
-------------------------------------------------------------,X,P)
B*I + C*X + D
4*E *I
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
E *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 - 1116619298080.0*P + 11431916240900.0*P
10.0*P
- 23668120508900.0)/(E
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 4 3 2 2 2 2 3
(K*( - ---*P + ---*A))/(P*(P - 4*P *A + 6*P *A + 2*P *K - 4*P*A
3 3
2 4 2 2 4
- 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(2*PI - 3)
---------------
P*PI
E
laplace(e**(-b*x)*delta(x-tau), x, p);
1
----------------
P*TAU + B*TAU
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
----------------
P*TAU + B*TAU
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
- (A - P)
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
2 2
1 2*A*I*T 2 2*A*I*T
- P*A*I + P*K - ---*E *A + E *A*I*K
2
1 2*A*I*T 2 1 2 1 2 P*T + A*I*T
+ ---*E *K + ---*A + A*I*K - ---*K ))/(E *I
2 2 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(X + TAU),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(X + TAU),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(X + TAU),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);
*** Laplace for LOG(X) not known - try ON LMON
1
LAPLACE( - ---*LOG(X)*A,X,P)
4
laplace(-log(x),x,p);
*** Laplace for LOG(X) not known - try ON LMON
LAPLACE( - LOG(X),X,P)
laplace(a*log(x)*e**(k*x), x, p);
*** Laplace for LOG(X) not known - try ON LMON
K*X
LAPLACE(E *LOG(X)*A,X,P)
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);
*** Laplace for DF(F(X),X) not known - try ON LMON
1
LAPLACE( - ---*DF(F(X),X)*A*C,X,P)
3
laplace(1/2*a*df(-2/3*f(x)*c,x,4), x,p);
*** Laplace for DF(F(X),X,4) not known - try ON LMON
1
LAPLACE( - ---*DF(F(X),X,4)*A*C,X,P)
3
laplace(1/2*a*e**(k*x)*df(-2/3*f(x)*c,x,2), x,p);
*** Laplace for DF(F(X),X,2) not known - try ON LMON
K*X
- E *DF(F(X),X,2)*A*C
LAPLACE(--------------------------,X,P)
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);
*** Laplace for DF(F(X),X) not known - try ON LMON
1
LAPLACE( - ---*DF(F(X),X)*A*C,X,P)
3
laplace(1/2*a*df(-2/3*f(x)*c,x,4), x,p);
*** Laplace for DF(F(X),X,4) not known - try ON LMON
1
LAPLACE( - ---*DF(F(X),X,4)*A*C,X,P)
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
C*(3/16*(4*I + 5*K - P) + 3/16*(4*I - 5*K + P)
-4 -4 -4
- 3/4*(2*I + 5*K - P) - 3/4*(2*I - 5*K + P) + 9/8*(5*K - P)
)
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
C*(3/16*(4*I + 5*K - P) + 3/16*(4*I - 5*K + P)
-4 -4 -4
- 3/4*(2*I + 5*K - P) - 3/4*(2*I - 5*K + P) + 9/8*(5*K - P)
)
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 - SIN(X*A - B)*COS(X*C + D) not known - try ON LMON
LAPLACE( - SIN(A*X - B)*COS(C*X + D),X,P)
laplace(x**(-5/2),x,p);
*** Laplace for X**( - 5/2) not known - try ON LMON
- 5/2
LAPLACE(X ,X,P)
% 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
-1 A*X
P *LAPLACE(E *SIN(K*X - TAU)*X,X,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
2 -1
LAPLACE(3*(X - 5*X + 6) *X,X,P)
laplace(1/sin(x),x,p);
*** Laplace for SIN(X)**(-1) not known - try ON LMON
-1
LAPLACE(SIN(X) ,X,P)
% But ...
laplace(x/sin(-3*a**2),x,p);
-2 2 -1
- 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);
*** Invlap for P**(-1) not known
-1
INVLAP(P ,P,T)
invlap(1/p**3, p, t);
*** Invlap for P**(-3) not known
-3
INVLAP(P ,P,T)
invlap(1/(p-a), p, t);
T*A
E
invlap(1/(2*p-a),p,t);
1/2*T*A
1/2*E
invlap(1/(p/2-a),p,t);
2*T*A
2*E
invlap(e**(-k*p)/(p-a), p, t);
T*A - A*K
E
invlap(b**(-k*p)/(p-a), p, t);
T*A - LOG(B)*A*K
E
invlap(1/(p-a)**3, p, t);
T*A 2
1/2*E *T
invlap(1/(c*p-a)**3, p, t);
-1
T*A*C 2 -3
1/2*E *T *C
invlap(1/(p/c-a)**3, p, t);
T*A*C 2 3
1/2*E *T *C
invlap((c*p-a)**(-1)/(c*p-a)**2, p, t);
-1
T*A*C 2 -3
1/2*E *T *C
invlap(c/((p/c-a)**2*(p-a*c)), p, t);
T*A*C 2 3
1/2*E *T *C
invlap(1/(p*(p-a)), p, t);
*** Invlap for P**(-1)*(P - A)**(-1) not known
-1 -1
INVLAP( - (A - P) *P ,P,T)
invlap(c/((p-a)*(p-b)), p, t);
-1 T*A T*B
(A - B) *C*(E - E )
invlap(p/((p-a)*(p-b)), p, t);
*** Invlap for P*(P - A)**(-1)*(P - B)**(-1) not known
-1 -1
INVLAP((A - P) *(B - P) *P,P,T)
off mcd;
invlap((p+d)/(p*(p-a)), p, t);
*** Invlap for P**(-1)*(P - A)**(-1)*(P + D) not known
-1 -1 -1
INVLAP( - (A - P) *D*P - (A - P) ,P,T)
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);
*** Invlap for E**( - P*K)*P**(-1)*(P + 1)**(-1) not known
- K*P -1 -1
INVLAP(E *(P + 1) *P ,P,T)
on mcd;
off exp;
invlap(c/(p*(p+a)**2), p, t);
*** Invlap for P**(-1)*(P + A)**(-2)*C not known
C
INVLAP(------------,P,T)
2
(A + P) *P
on exp;
invlap(1, p, t);
DELTA(T)
invlap(c1*p/c2, p, t);
*** Invlap for P*C1*C2**(-1) not known
-1
INVLAP(P*C1*C2 ,P,T)
invlap(p/(p-a), p, t);
*** Invlap for P*(P - A)**(-1) not known
-1
INVLAP( - (A - P) *P,P,T)
invlap(c*p**2, p, t);
*** Invlap for P**2*C not known
2
INVLAP(C*P ,P,T)
invlap(p**2*e**(-a*p)*c, p, t);
*** Invlap for E**( - P*A)*P**2*C not known
- A*P 2
INVLAP(E *C*P ,P,T)
off mcd;
invlap(e**(-a*p)*(1/p**2-p/(p-1))+c/p, p, t);
*** Invlap for P**(-1)*C not known
*** Invlap for - E**( - P*A)*P**(-2)*(P**3*(P - 1)**(-1) - 1) not known
- A*P -1 - A*P -2 -1
INVLAP( - E *(P - 1) *P + E *P ,P,T) + INVLAP(C*P ,P,T)
on mcd;
invlap(a*p**2-2*p+1, p, x);
*** Invlap for P**2*A not known
*** Invlap for P not known
2
INVLAP( - 2*P,P,X) + INVLAP(A*P ,P,X) + DELTA(X)
% P to non-integer power in denominator - i.e. gamma-function case.
invlap(1/sqrt(p), p, t);
1
------------------
SQRT(PI)*SQRT(T)
invlap(1/sqrt(p-a), p, t);
T*A
E
------------------
SQRT(PI)*SQRT(T)
invlap(c/(p*sqrt(p)), p, t);
*** Invlap for P**( - 1/2)*P**(-1)*C not known
- 1/2 -1
INVLAP(P *C*P ,P,T)
invlap(c*sqrt(p)/p**2, p, t);
*** Invlap for SQRT(P)*P**(-2)*C not known
-2
INVLAP(SQRT(P)*C*P ,P,T)
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);
1 (T*A)/2
---*SQRT(T)*E *SQRT(2)*C
2
--------------------------------
SQRT(PI)
invlap(c/p**(7/2), p, t);
8 2
----*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);
1
---------------------------------------
- T*A + A*K
SQRT(PI)*E *SQRT( - K + T)
% Images that give elementary object functions.
% Use of new switches lmon, lhyp.
invlap(k/(p**2+k**2), p, t);
1 2*T*I*K 1
---*E - ---
2 2
--------------------
T*I*K
E *I
% This is made more readable by :
on ltrig;
invlap(k/(p**2+k**2), p, t);
SIN(K*T)
invlap(p/(p**2+1), p, t);
*** Invlap for P*(P**2 + 1)**(-1) not known
P
INVLAP(--------,P,T)
2
P + 1
invlap((p**2-a**2)/(p**2+a**2)**2, p, t);
T*COS(A*T)
invlap(p/(p**2+a**2)**2, p, t);
*** Invlap for P*(P**2 + A**2)**(-2) not known
P
INVLAP(-------------------,P,T)
4 2 2 4
A + 2*A *P + P
invlap((p-a)/((p-a)**2+b**2), p, t);
T*A
E *COS(B*T)
off ltrig;
on lhyp;
invlap(s/(s**2-k**2), s, t);
*** Invlap for S*(S**2 - K**2)**(-1) not known
- S
INVLAP(---------,S,T)
2 2
K - S
invlap(e**(-tau/k*p)*p/(p**2-k**2), p, t);
*** Invlap for E**( - P*K**(-1)*TAU)*P*(P**2 - K**2)**(-1) not known
- P
INVLAP(-------------------------------,P,T)
(P*TAU)/K 2 (P*TAU)/K 2
E *K - E *P
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(K*T)*(COSH(A*T) + SINH(A*T))
off lhyp;
on ltrig;
invlap(e**(-tau/k*p)*k/(p**2+k**2), p, t);
1 2*T*I*K 1 2*I*TAU
---*E - ---*E
2 2
-----------------------------
T*I*K + I*TAU
E *I
off ltrig;
% In such situations use the default switches:
invlap(k/((p-a)**2-k**2), p, t);
T*A 1 2*T*K 1
E *(---*E - ---)
2 2
-------------------------
T*K
E
% i.e. e**(a*t)*cosh(k*t).
invlap(e**(-tau/k*p)*k/(p**2+k**2), p, t);
1 2*T*I*K 1 2*I*TAU
---*E - ---*E
2 2
-----------------------------
T*I*K + I*TAU
E *I
% i.e. sin(k*t-tau).
% More complicated examples.
off exp,mcd;
invlap((p+d)/(p**2*(p-a)), p, t);
*** Invlap for P**(-2)*(P - A)**(-1)*(P + D) not known
-1 -2
INVLAP( - (A - P) *(D + P)*P ,P,T)
invlap(e**(-tau/k*p)*c/(p*(p-a)**2), p, t);
*** Invlap for E**( - P*K**(-1)*TAU)*P**(-1)*(P - A)**(-2)*C not known
-1
- K *P*TAU -2 -1
INVLAP(E *(A - P) *C*P ,P,T)
invlap(1/((p-a)*(p-b)*(p-c)), p, t);
T*A 2 -1 T*B 2 -1
E *(A - A*B - A*C + B*C) - E *(A*B - A*C - B + B*C)
T*C 2 -1
+ E *(A*B - A*C - B*C + C )
invlap((p**2+g*p+d)/(p*(p-a)**2), p, t);
*** Invlap for P**(-1)*(P**2 + P*G + D)*(P - A)**(-2) not known
-2 2 -1
INVLAP((A - P) *(D + G*P + P )*P ,P,T)
on exp,mcd;
invlap(k*c**(-b*p)/((p-a)**2+k**2), p, t);
1 2*B*I*K 1 2*T*I*K
- ---*C + ---*E
2 2
-----------------------------------------------
- T*A + T*I*K + LOG(C)*A*B + LOG(C)*B*I*K
E *I
on ltrig;
invlap(c/(p**2*(p**2+a**2)), p, t);
*** Invlap for P**(-2)*(P**2 + A**2)**(-1)*C not known
C
INVLAP(------------,P,T)
2 2 4
A *P + P
invlap(1/(p**2-p+1), p, t);
T/2 1
2*E *SIN(---*SQRT(3)*T)
2
---------------------------
SQRT(3)
invlap(1/(p**2-p+1)**2, p, t);
T/2 1 1
2*E *( - 3*T*COS(---*SQRT(3)*T) + 2*SQRT(3)*SIN(---*SQRT(3)*T))
2 2
-------------------------------------------------------------------
9
invlap(2*a**2/(p*(p**2+4*a**2)), p, t);
*** Invlap for 2*P**(-1)*(P**2 + 4*A**2)**(-1)*A**2 not known
2
2*A
INVLAP(-------------,P,T)
2 3
4*A *P + P
% 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);
*** Invlap for 2*P**(-1)*(P**2 + 4*A**2)**(-1)*A**2 not known
2
2*A
INVLAP(-------------,P,T)
2 3
4*A *P + P
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);
*** Invlap for P**(-1)*(P**2 - 2*A**2)*(P**2 - 4*A**2)**(-1) not known
2 2
2*A - P
INVLAP(-------------,P,T)
2 3
4*A *P - P
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);
51000 (33333*T)/10000 51000 4*T
- -------*E + -------*E
6667 6667
on rounded;
invlap(2.55/((0.5*p-2.0)*(p-3.3333)), p, t);
4.0*T 3.3333*T
7.64961751912*E - 7.64961751912*E
invlap(1.5/sqrt(p-0.5), p, t);
0.5*T
1.5*E
-----------------
0.5
T *GAMMA(0.5)
invlap(2.75*p**2-0.5*p+e**(-0.9*p)/p, p, t);
*** Invlap for P**2 not known
*** Invlap for P not known
*** Invlap for E**( - 0.9*P)*P**(-1) not known
- 0.9*P -1
INVLAP( - 0.5*P,P,T) + INVLAP(E *P ,P,T)
2
+ INVLAP(2.75*P ,P,T)
invlap(1/(2.0*p-3.0)**3, p, t);
1.5*T 2
0.0625*E *T
invlap(1/(2.0*p-3.0)**(3/2), p, t);
0.5 1.5*T
0.353553390593*T *E
----------------------------
GAMMA(1.5)
invlap(1/(p**2-5.0*p+6), p, t);
3.0*T 2.0*T
E - E
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);
*** Invlap for - 1/2*P**(-1)*LOG(P*GAM) not known
1 -1
INVLAP( - ---*LOG(P*GAM)*P ,P,T)
2
invlap(-e**(-a*p)*log(gam*p)/(c*p), p, t);
*** Invlap for - E**( - P*A)*P**(-1)*LOG(P*GAM)*C**(-1) not known
- A*P -1 -1
INVLAP( - E *LOG(P*GAM)*C *P ,P,T)
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);
*** Invlap for C**( - P*K)*P**(-2)*(P**2 + P*G + D)*(P - A)**(-3) not known
- K*P -3 2 -2
INVLAP( - C *(A - P) *(D + G*P + P )*P ,P,T)
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(A*T) + SINH(A*T)
-------------------------
3
2*A
invlap(1/((b-3)*p**4-a**4*(2+b-5)), p, t);
- SIN(A*T) + SINH(A*T)
-------------------------
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);
*** Invlap for (1/8*P**3 - 9/4*P**2 + 27/2*P - 27)**(-2)*C not known
1 3 9 2 27 -2
INVLAP((---*P - ---*P + ----*P - 27) *C,P,T)
8 4 2
invlap(c/(p/2-3)**6,p,t);
8 6*T 5
----*E *T *C
15
off exp;
a:=(p/2-3)**6;
6
(P - 6)
A := ----------
64
on exp;
invlap(c/a, p, t);
8 6*T 5
----*E *T *C
15
clear a;
% The following two examples are the same :
invlap(c/(p**4+2*p**2+1)**2, p, t);
*** Invlap for (P**4 + 2*P**2 + 1)**(-2)*C not known
4 2 -2
INVLAP((P + 2*P + 1) *C,P,T)
invlap(c/((p-i)**4*(p+i)**4),p,t);
1 2*T*I 3 1 3 1 2*T*I 2 1 2
(C*(----*E *T *I + ----*T *I - ----*E *T + ----*T
96 96 16 16
5 2*T*I 5 5 2*T*I 5 T*I
- ----*E *T*I - ----*T*I + ----*E - ----))/(E *I)
32 32 32 32
% The following three examples are the same :
invlap(e**(-k*p)/(2*p-3)**6, p, t);
(3*T)/2 1 5 4 3 2 2 3 4 1 5
E *(---*T - T *K + 2*T *K - 2*T *K + T*K - ---*K )
5 5
--------------------------------------------------------------
(3*K)/2
1536*E
invlap(e**(-k*p)/(4*p**2-12*p+9)**3, p, t);
(3*T)/2 1 5 4 3 2 2 3 4 1 5
E *(---*T - T *K + 2*T *K - 2*T *K + T*K - ---*K )
5 5
--------------------------------------------------------------
(3*K)/2
1536*E
invlap(e**(-k*p)/(8*p**3-36*p**2+54*p-27)**2, p, t);
*** Invlap for E**( - P*K)*(8*P**3 - 36*P**2 + 54*P - 27)**(-2) not known
- K*P 3 2 -2
INVLAP(E *(8*P - 36*P + 54*P - 27) ,P,T)
% Error messages.
invlap(e**(a*p)/p, p, t);
*** Invlap for E**(P*A)*P**(-1) not known
A*P -1
INVLAP(E *P ,P,T)
invlap(c*p*sqrt(p), p, t);
*** Invlap for SQRT(P)*P*C 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
3 2 -1
INVLAP((A*P + B*P + C*P + D) ,P,T)
invlap(1/(p**2-p*sin(p)+a**2),p,t);
*** Invlap for (P**2 - P*SIN(P) + A**2)**(-1) not known
2 2 -1
INVLAP( - (SIN(P)*P - A - P ) ,P,T)
on rounded;
invlap(1/(p**3-1), p, t);
*** Invlap for (P**3 - 1)**(-1) not known
3 -1
INVLAP((P - 1) ,P,T)
off rounded;
% Severe errors:
%invlap(1/(p**2+1), p+1, sin(t) );
%invlap(p/(p+1)**2, sin(p), t);
end;
Quitting