@@ -1,1818 +1,1818 @@ -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(,,0,). - -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) +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(,,0,). + +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)