Artifact 0454e0d377d791c65c209bc70ef1b4b6f85ca9d57018d7b5c0ce748a3d85c105:
- File
r34/lib/laplace.log
— part of check-in
[f2fda60abd]
at
2011-09-02 18:13:33
on branch master
— Some historical releases purely for archival purposes
git-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1375 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: arthurcnorman@users.sourceforge.net, size: 32845) [annotate] [blame] [check-ins using] [more...]
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