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r36/xlog/ZTRANS.LOG
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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... % test file for ztrans package % operator f; operator g; operator h; % Examples for Z transformation ztrans(1,n,z); z ------- z - 1 ztrans(a,n,z); a*z ------- z - 1 ztrans((-1)^n,n,z); z ------- z + 1 ztrans(n,n,z); z -------------- 2 z - 2*z + 1 ztrans(n^2,n,z); z*(z + 1) --------------------- 3 2 z - 3*z + 3*z - 1 ztrans(n^k,n,z); k ztrans(n ,n,z) % should be output=input ztrans((-1)^n*n^2,n,z); z*( - z + 1) --------------------- 3 2 z + 3*z + 3*z + 1 ztrans(binomial(n,m),n,z); z ------------------ m (z - 1) *(z - 1) ztrans((-1)^n*binomial(n,m),n,z); z --------------------- m ( - z - 1) *(z + 1) ztrans(binomial(n+k,m),n,z); k z *z ------------------ m (z - 1) *(z - 1) ztrans(a^n,n,z); - z ------- a - z ztrans(a^(n-1),n,z); - z ----------- a*(a - z) ztrans(a^(n+k),n,z); k - a *z --------- a - z ztrans((-1)^n*a^n,n,z); z ------- a + z ztrans(1-a^n,n,z); z*(a - 1) ------------------ 2 a*z - a - z + z ztrans(n*a^n,n,z); a*z ----------------- 2 2 a - 2*a*z + z ztrans(n^3*a^n,n,z); 2 2 a*z*(a + 4*a*z + z ) ------------------------------------- 4 3 2 2 3 4 a - 4*a *z + 6*a *z - 4*a*z + z ztrans(binomial(n,m)*a^n,n,z); m - a *z --------------------- m ( - a + z) *(a - z) ztrans(1/(n+1),n,z); z log(-------)*z z - 1 ztrans(1/(n+2),n,z); z z*(log(-------)*z - 1) z - 1 ztrans((-1)^(n)/(n+1),n,z); z - log(-------)*z z + 1 ztrans((-1)^(n)/(n+2),n,z); z z*(log(-------)*z + 1) z + 1 ztrans(a^(n-1)/(n+1),n,z); - z log(-------)*z a - z ---------------- 2 a ztrans(a^(n+k)/(n+1),n,z); k - z a *log(-------)*z a - z ------------------- a ztrans(a^n/factorial(n),n,z); a/z e ztrans((n+1)*a^n/factorial(n),n,z); a/z e *(a + z) -------------- z ztrans(1/factorial(n-1),n,z); 1 ***** ERROR: zero divisor in sum(---------------------,n,0,infinity) n z *factorial(n - 1) % ERROR message o.k. ztrans((-1)^n/factorial(2*n+1),n,z); 1 sqrt(z)*sin(---------) sqrt(z) ztrans((-1)^n/factorial(2*n),n,z); 1 cos(---------) sqrt(z) ztrans(1/factorial(2*n+1),n,z); 1 sqrt(z)*sinh(---------) sqrt(z) ztrans(1/factorial(2*n-1),n,z); 1 ztrans(--------------------,n,z) factorial(2*n - 1) ztrans(1/factorial(2*n+3),n,z); 1 z*(sqrt(z)*sinh(---------) - 1) sqrt(z) ztrans(1/factorial(2*n),n,z); 1 cosh(---------) sqrt(z) ztrans(1/factorial(2*n+2),n,z); 1 z*(cosh(---------) - 1) sqrt(z) ztrans(a^n/factorial(2*n+1),n,z); sqrt(a) sqrt(z)*sinh(---------) sqrt(z) ------------------------- sqrt(a) ztrans(a^n/factorial(2*n),n,z); sqrt(a) cosh(---------) sqrt(z) ztrans(e^(a*n),n,z); - z -------- a e - z ztrans(e^(a*(n+k)),n,z); a*k - e *z ----------- a e - z ztrans(sinh(a*n),n,z); - sinh(a)*z ---------------------- 2 2*cosh(a)*z - z - 1 ztrans(cosh(a*n),n,z); z*(cosh(a) - z) ---------------------- 2 2*cosh(a)*z - z - 1 ztrans(sinh(a*n+p),n,z); - z*(sinh(a - p) + sinh(p)*z) -------------------------------- 2 2*cosh(a)*z - z - 1 ztrans(cosh(a*n+p),n,z); z*(cosh(a - p) - cosh(p)*z) ----------------------------- 2 2*cosh(a)*z - z - 1 ztrans(a^n*sinh(a*n),n,z); - sinh(a)*a*z ------------------------- 2 2 2*cosh(a)*a*z - a - z ztrans(a^n*cosh(a*n),n,z); z*(cosh(a)*a - z) ------------------------- 2 2 2*cosh(a)*a*z - a - z ztrans(n*sinh(a*n),n,z); 2 sinh(a)*z*(z - 1) ------------------------------------------------------------ 2 2 3 4 2 4*cosh(a) *z - 4*cosh(a)*z - 4*cosh(a)*z + z + 2*z + 1 ztrans(n*cosh(a*n),n,z); 2 z*(cosh(a)*z + cosh(a) - 2*z) ------------------------------------------------------------ 2 2 3 4 2 4*cosh(a) *z - 4*cosh(a)*z - 4*cosh(a)*z + z + 2*z + 1 ztrans(n^2*a^n*sinh(b*n),n,z); 2 4 2 2 2 4 3 3 6 (sinh(b)*a*z*( - 4*cosh(b) *a *z - 4*cosh(b) *a *z + 16*cosh(b)*a *z + a 4 2 2 4 6 4 4 4 3 5 3 - 5*a *z - 5*a *z + z ))/(16*cosh(b) *a *z - 32*cosh(b) *a *z 3 3 5 2 6 2 2 4 4 - 32*cosh(b) *a *z + 24*cosh(b) *a *z + 48*cosh(b) *a *z 2 2 6 7 5 3 3 5 + 24*cosh(b) *a *z - 8*cosh(b)*a *z - 24*cosh(b)*a *z - 24*cosh(b)*a *z 7 8 6 2 4 4 2 6 8 - 8*cosh(b)*a*z + a + 4*a *z + 6*a *z + 4*a *z + z ) ztrans(sin(b*n),n,z); - sin(b)*z --------------------- 2 2*cos(b)*z - z - 1 ztrans(cos(b*n),n,z); z*(cos(b) - z) --------------------- 2 2*cos(b)*z - z - 1 ztrans(sin(b*n+p),n,z); - z*(sin(b - p) + sin(p)*z) ------------------------------ 2 2*cos(b)*z - z - 1 ztrans(cos(b*n+p),n,z); z*(cos(b - p) - cos(p)*z) --------------------------- 2 2*cos(b)*z - z - 1 ztrans(e^(a*n)*sin(b*n),n,z); a - e *sin(b)*z --------------------------- a 2*a 2 2*e *cos(b)*z - e - z ztrans(e^(a*n)*cos(b*n),n,z); a z*(e *cos(b) - z) --------------------------- a 2*a 2 2*e *cos(b)*z - e - z ztrans((-1)^n*e^(a*n)*sin(b*n),n,z); a - e *sin(b)*z --------------------------- a 2*a 2 2*e *cos(b)*z + e + z ztrans((-1)^n*e^(a*n)*cos(b*n),n,z); a z*(e *cos(b) + z) --------------------------- a 2*a 2 2*e *cos(b)*z + e + z ztrans(n*sin(b*n),n,z); 2 sin(b)*z*(z - 1) --------------------------------------------------------- 2 2 3 4 2 4*cos(b) *z - 4*cos(b)*z - 4*cos(b)*z + z + 2*z + 1 ztrans(n*cos(b*n),n,z); 2 z*(cos(b)*z + cos(b) - 2*z) --------------------------------------------------------- 2 2 3 4 2 4*cos(b) *z - 4*cos(b)*z - 4*cos(b)*z + z + 2*z + 1 ztrans(n^2*a^n*sin(b*n),n,z); 2 4 2 2 2 4 3 3 6 (sin(b)*a*z*( - 4*cos(b) *a *z - 4*cos(b) *a *z + 16*cos(b)*a *z + a 4 2 2 4 6 4 4 4 3 5 3 - 5*a *z - 5*a *z + z ))/(16*cos(b) *a *z - 32*cos(b) *a *z 3 3 5 2 6 2 2 4 4 2 2 6 - 32*cos(b) *a *z + 24*cos(b) *a *z + 48*cos(b) *a *z + 24*cos(b) *a *z 7 5 3 3 5 7 8 - 8*cos(b)*a *z - 24*cos(b)*a *z - 24*cos(b)*a *z - 8*cos(b)*a*z + a 6 2 4 4 2 6 8 + 4*a *z + 6*a *z + 4*a *z + z ) ztrans(cos(b*(n+1))/(n+1),n,z); z log(------------------------------)*z 2 sqrt( - 2*cos(b)*z + z + 1) ztrans(sin(b*(n+1))/(n+1),n,z); sin(b) - atan(------------)*z cos(b) - z ztrans(cos(b*(n+2))/(n+2),n,z); z z*( - cos(b) + log(------------------------------)*z) 2 sqrt( - 2*cos(b)*z + z + 1) ztrans((-1)^(n)*cos(b*(n+1))/(n+1),n,z); 2 3 sqrt(2*cos(b)*z + z + 1) - log(----------------------------)*z sqrt(z) ztrans((-1)^(n)*sin(b*(n+1))/(n+1),n,z); sin(b) atan(------------)*z cos(b) + z ztrans(cos(b*n)/factorial(n),n,z); cos(b)/z sin(b) e *cos(--------) z ztrans(sin(b*n)/factorial(n),n,z); cos(b)/z sin(b) e *sin(--------) z ztrans(a*f(n)+b*g(n)+c*h(n),n,z); ztrans(f(n),n,z)*a + ztrans(g(n),n,z)*b + ztrans(h(n),n,z)*c ztrans(sum(f(k)*g(n-k),k,0,n),n,z); ztrans(f(n),n,z)*ztrans(g(n),n,z) ztrans(sum(f(k),k,0,n),n,z); ztrans(f(n),n,z)*z -------------------- z - 1 ztrans(sum(f(k),k,-2,n),n,z); 2 (z*( - f(-1)*z + f(-1) - f(-2)*z + f(-2) + ztrans(f(n - 2),n,z) 2 + ztrans(f(n - 2),n,z)*z - ztrans(f(n - 2),n,z)))/(z - 1) ztrans(sum(f(k),k,3,n),n,z); 2 2 - f(2) - f(1)*z - f(0)*z + ztrans(f(n),n,z)*z -------------------------------------------------- z*(z - 1) ztrans(sum(f(k),k,0,n+2),n,z); 2 2 (z*( - f(1)*z + f(1) - f(0)*z + f(0) + ztrans(f(n),n,z) + ztrans(f(n),n,z)*z - ztrans(f(n),n,z)))/(z - 1) ztrans(sum(f(k),k,0,n-3),n,z); 2 2 ztrans(f(n),n,z)*z - ztrans(f(n),n,z)*z + ztrans(f(n),n,z) -------------------------------------------------------------- 2 z *(z - 1) ztrans(sum(f(k),k,-2,n+3),n,z); 2 3 4 (z*( - f(2)*z + f(2) - f(1)*z + f(1) - f(0)*z + f(0) - f(-1)*z + f(-1) 5 5 - f(-2)*z + f(-2) + ztrans(f(n - 2),n,z) + ztrans(f(n - 2),n,z)*z - ztrans(f(n - 2),n,z)))/(z - 1) ztrans(sum(1/factorial(k),k,0,n),n,z); 1/z e *z -------- z - 1 ztrans(sum(1/factorial(k+2),k,0,n),n,z); 2 1/z z *(e *z - z - 1) --------------------- z - 1 ztrans(n^2*sum(1/factorial(k),k,0,n),n,z); 1/z 3 2 e *(2*z + 2*z - 3*z + 1) ------------------------------ 3 2 z*(z - 3*z + 3*z - 1) ztrans(sum(n^2/factorial(k),k,0,n),n,z); 1/z 3 2 e *(2*z + 2*z - 3*z + 1) ------------------------------ 3 2 z*(z - 3*z + 3*z - 1) ztrans(sum(1/k,k,0,n),n,z); 1 ***** ERROR: zero divisor in sum(------,n,0,infinity) n z *n % ERROR o.k. ztrans(sum(1/(k+1),k,0,n),n,z); z 2 log(-------)*z z - 1 ----------------- z - 1 ztrans(sum(1/(k+3),k,0,n),n,z); 2 z 2 z *(2*log(-------)*z - 2*z - 1) z - 1 ---------------------------------- 2*(z - 1) ztrans(f(n+k),n,z); ztrans(f(k + n),n,z) % output=input ztrans(f(n+2),n,z); z*( - f(1) - f(0)*z + ztrans(f(n),n,z)*z) ztrans(f(n-k),n,z); ztrans(f( - k + n),n,z) % output=input ztrans(f(n-3),n,z); ztrans(f(n - 3),n,z) % output=input ztrans(a^n*f(n),n,z); z ztrans(f(n),n,---) a ztrans(n*f(n),n,z); - df(ztrans(f(n),n,z),z)*z ztrans(1/a^n,n,z); a*z --------- a*z - 1 ztrans(1/a^(n+1),n,z); z --------- a*z - 1 ztrans(1/a^(n-1),n,z); 2 a *z --------- a*z - 1 ztrans(2*n+n^2-3/4*n^3,n,x); 2 x*(9*x - 28*x + 1) -------------------------------- 4 3 2 4*(x - 4*x + 6*x - 4*x + 1) ztrans(n^2*cos(n*x),n,z); 3 4 3 2 6 4 2 (z*( - 4*cos(x) *z + 4*cos(x) *z + cos(x)*z + 9*cos(x)*z - 9*cos(x)*z 5 4 4 3 5 3 3 - cos(x) - 4*z + 4*z))/(16*cos(x) *z - 32*cos(x) *z - 32*cos(x) *z 2 6 2 4 2 2 7 5 + 24*cos(x) *z + 48*cos(x) *z + 24*cos(x) *z - 8*cos(x)*z - 24*cos(x)*z 3 8 6 4 2 - 24*cos(x)*z - 8*cos(x)*z + z + 4*z + 6*z + 4*z + 1) ztrans((1+n)^2*f(n),n,z); 2 df(ztrans(f(n),n,z),z,2)*z - df(ztrans(f(n),n,z),z)*z + ztrans(f(n),n,z) ztrans(n^2*f(n),n,z); z*(df(ztrans(f(n),n,z),z,2)*z + df(ztrans(f(n),n,z),z)) ztrans(n/factorial(n),n,z); 1/z e ------ z ztrans(n^2/factorial(n),n,z); 1/z e *(z + 1) -------------- 2 z ztrans(a^n/factorial(n),n,z); a/z e ztrans(1/(a^n*factorial(n)),n,z); 1/(a*z) e ztrans(sum(f(k)*g(n-k),k,0,n),n,z); ztrans(f(n),n,z)*ztrans(g(n),n,z) ztrans(sum(f(k),k,0,n-1),n,z); ztrans(f(n),n,z) ------------------ z - 1 ztrans(sum(f(k),k,0,n),n,z); ztrans(f(n),n,z)*z -------------------- z - 1 ztrans(sum(1/factorial(k),k,0,n),n,z); 1/z e *z -------- z - 1 ztrans(sum(k/factorial(k),k,0,n),n,z); 1/z e ------- z - 1 ztrans(sum(a^k*k^2/factorial(k),k,0,n),n,z); a/z e *a*(a + z) ---------------- z*(z - 1) ztrans(a^n*f(n),n,z); z ztrans(f(n),n,---) a ztrans(binomial(n,k),n,z); z ------------------ k (z - 1) *(z - 1) ztrans(1/(n+1),n,z); z log(-------)*z z - 1 ztrans(n/factorial(2*n+1),n,z); 1 1 cosh(---------) - sqrt(z)*sinh(---------) sqrt(z) sqrt(z) ------------------------------------------- 2 ztrans(a^n*sin(n*x+y),n,z); - z*(sin(x - y)*a + sin(y)*z) -------------------------------- 2 2 2*cos(x)*a*z - a - z ztrans(n^3*sin(n*x+y),n,z); 3 4 2 4 2 2 (z*(8*cos(x) *sin(y)*z + 4*cos(x) *sin(x - y)*z - 4*cos(x) *sin(x - y)*z 2 5 2 3 5 + 16*cos(x) *sin(y)*z - 16*cos(x) *sin(y)*z + 8*cos(x)*sin(x - y)*z 6 4 - 8*cos(x)*sin(x - y)*z + 2*cos(x)*sin(y)*z - 36*cos(x)*sin(y)*z 2 6 4 2 + 10*cos(x)*sin(y)*z + sin(x - y)*z - 23*sin(x - y)*z + 23*sin(x - y)*z 5 3 4 4 - sin(x - y) - 8*sin(y)*z + 32*sin(y)*z - 8*sin(y)*z))/(16*cos(x) *z 3 5 3 3 2 6 2 4 - 32*cos(x) *z - 32*cos(x) *z + 24*cos(x) *z + 48*cos(x) *z 2 2 7 5 3 + 24*cos(x) *z - 8*cos(x)*z - 24*cos(x)*z - 24*cos(x)*z - 8*cos(x)*z 8 6 4 2 + z + 4*z + 6*z + 4*z + 1) ztrans((n+1)/factorial(n),n,z); 1/z e *(z + 1) -------------- z ztrans(factorial(n)/(factorial(k)*factorial(n-k)),n,z); z ------------------ k (z - 1) *(z - 1) % Examples for inverse Z transformation invztrans(z/(z-1),z,n); 1 invztrans(z/(z+1),z,n); n ( - 1) invztrans(z/(z-1)^2,z,n); n invztrans(z*(z+1)/(z-1)^3,z,n); 2 n invztrans(z/(z-1)^m,z,n); binomial(n,m - 1) % invztrans(z/(z-1)^(m+1),z,n); % not yet supported invztrans(z/(z-1)^4,z,n); 2 n*(n - 3*n + 2) ------------------ 6 invztrans((-1)^m*z/(z+1)^m,z,n); m ( - 1) *z invztrans(-----------,z,n) m (z + 1) % not yet supported invztrans(z/(z+1)^4,z,n); n 2 ( - 1) *n*( - n + 3*n - 2) ----------------------------- 6 % invztrans(z^(k+1)/(z-1)^(m+1),z,n); % not yet supported invztrans(z^4/(z-1)^m,z,n); binomial(n + 3,m - 1) % invztrans(z^4/(z-1)^(m+1),z,n); % not yet supported % invztrans(z^4/(z-1)^m,z,n); % not yet supported % invztrans(z^(k+1)/(z-1)^5,z,n); % not yet supported invztrans(z^3/(z-a)^4,z,n); n 2 a *n*(n + 3*n + 2) --------------------- 6*a invztrans(z/(z-a),z,n); n a invztrans(z/(z+a),z,n); n ( - a) invztrans(z*(1-a)/((z-1)*(z-a)),z,n); n - a + 1 invztrans(z*a/(z-a)^2,z,n); n a *n invztrans(z*3/(z-3)^2,z,n); n 3 *n % invztrans(a^m*z/(z-a)^(m+1),z,n); % not yet supported % invztrans(a^m*z/(z-a)^m,z,n); % not yet supported % invztrans(4^m*z/(z-4)^(m+1),z,n); % not yet supported invztrans(a^3*z/(z-a)^5,z,n); n 3 2 a *n*(n - 6*n + 11*n - 6) ----------------------------- 24*a invztrans(z*log(z/(z-1)),z,n); 1 ------- n + 1 invztrans(z*log(1+1/z),z,n); n ( - 1) --------- n + 1 invztrans(z*log(z/(z-a)),z,n); n a *a ------- n + 1 invztrans(e^(a/z),z,n); n a -------------- factorial(n) invztrans(e^(1/(a*z)),z,n); 1 ----------------- n a *factorial(n) invztrans((1+a/z)*e^(a/z),z,n); n a *(n + 1) -------------- factorial(n) invztrans(e^(a/z)*(a+z)/z,z,n); n a *(n + 1) -------------- factorial(n) invztrans(sqrt(z)*sin(1/sqrt(z)),z,n); n ( - 1) -------------------- factorial(2*n + 1) invztrans(cos(1/sqrt(z)),z,n); n ( - 1) ---------------- factorial(2*n) invztrans(sqrt(z)*sinh(1/sqrt(z)),z,n); 1 -------------------- factorial(2*n + 1) invztrans(cosh(1/sqrt(z)),z,n); 1 ---------------- factorial(2*n) invztrans(sqrt(z/a)*sinh(sqrt(a/z)),z,n); n a -------------------- factorial(2*n + 1) invztrans(cosh(sqrt(a/z)),z,n); n a ---------------- factorial(2*n) invztrans(z/(z-e^a),z,n); a*n e invztrans(z*sinh(a)/(z^2-2*z*cosh(a)+1),z,n); sinh(a*n) invztrans(z*(z-cosh(a))/(z^2-2*z*cosh(a)+1),z,n); cosh(a*n) invztrans(z*(z*sinh(p)+sinh(a-p))/(z^2-2*z*cosh(a)+1),z,n); cosh(a*n)*sinh(a)*sinh(p) + cosh(a)*sinh(a*n)*sinh(p) + sinh(a - p)*sinh(a*n) ------------------------------------------------------------------------------- sinh(a) % trigsimp(ws); % trigsimp(ws,combine); invztrans(z*(z*cosh(p)-cosh(a-p))/(z^2-2*z*cosh(a)+1),z,n); ( - cosh(a - p)*sinh(a*n) + cosh(a*n)*cosh(p)*sinh(a) + cosh(a)*cosh(p)*sinh(a*n))/sinh(a) % trigsimp(ws); % trigsimp(ws,combine); invztrans(a*z*sinh(a)/(z^2-2*a*z*cosh(a)+a^2),z,n); n a *sinh(a*n) invztrans(z*(z-a*cosh(a))/(z^2-2*a*z*cosh(a)+a^2),z,n); n a *cosh(a*n) invztrans(z*(z^2-1)*sinh(a)/(z^2-2*z*cosh(a)+1)^2,z,n); 2 sinh(a*n)*sinh(a) *n ---------------------- 2 cosh(a) - 1 % trigsimp(ws); invztrans(z*((z^2+1)*cosh(a)-2*z)/(z^2-2*z*cosh(a)+1)^2,z,n); cosh(a*n)*n invztrans(z*sin(b)/(z^2-2*z*cos(b)+1),z,n); sin(b*n) invztrans(z*(z-cos(b))/(z^2-2*z*cos(b)+1),z,n); cos(b*n) invztrans(z*(z*sin(p)+sin(b-p))/(z^2-2*z*cos(b)+1),z,n); cos(b*n)*sin(b)*sin(p) + cos(b)*sin(b*n)*sin(p) + sin(b - p)*sin(b*n) ----------------------------------------------------------------------- sin(b) % trigsimp(ws); % trigsimp(ws,combine); invztrans(z*(z*cos(p)-cos(b-p))/(z^2-2*z*cos(b)+1),z,n); - cos(b - p)*sin(b*n) + cos(b*n)*cos(p)*sin(b) + cos(b)*cos(p)*sin(b*n) -------------------------------------------------------------------------- sin(b) % trigsimp(ws); % trigsimp(ws,combine); invztrans(z*e^(a)*sin(b)/(z^2-2*z*e^a*cos(b)+e^(2*a)),z,n); a*n e *sin(b*n) invztrans(z*(z-e^a*cos(b))/(z^2-2*z*e^a*cos(b)+e^(2*a)),z,n); a*n e *cos(b*n) invztrans(-z*e^a*sin(b)/(z^2+2*z*e^a*cos(b)+e^(2*a)),z,n); a*n n e *( - 1) *sin(b*n) invztrans(z*(z+e^a*cos(b))/(z^2+2*z*e^a*cos(b)+e^(2*a)),z,n); a*n n e *( - 1) *cos(b*n) invztrans(z*(z^2-1)*sin(b)/(z^2-2*z*cos(b)+1)^2,z,n); 2 (sqrt(cos(b) - 1)*sin(b)*n 2 n 2 n *( - (cos(b) - sqrt(cos(b) - 1)) + (cos(b) + sqrt(cos(b) - 1)) ))/(2 2 *(cos(b) - 1)) % trigsimp(ws,expon); % trigsimp(ws,trig); invztrans(z*((z^2+1)*cos(b)-2*z)/(z^2-2*z*cos(b)+1)^2,z,n); 2 n 2 n n*((cos(b) - sqrt(cos(b) - 1)) + (cos(b) + sqrt(cos(b) - 1)) ) ------------------------------------------------------------------- 2 % trigsimp(ws,expon); % trigsimp(ws,trig); invztrans(z*log(z/sqrt(z^2-2*z*cos(b)+1)),z,n); cos(b*n + b) -------------- n + 1 invztrans(z*atan(sin(b)/(z-cos(b))),z,n); sin(b*n + b) -------------- n + 1 invztrans(z*log(sqrt(z^2+2*z*cos(b)+1)/z),z,n); n ( - 1) *cos(b*n + b) ---------------------- n + 1 invztrans(z*atan(sin(b)/(z+cos(b))),z,n); n ( - 1) *sin(b*n + b) ---------------------- n + 1 invztrans(cos(sin(b)/z)*e^(cos(b)/z),z,n); cos(b*n) -------------- factorial(n) invztrans(sin(sin(b)/z)*e^(cos(b)/z),z,n); sin(b*n) -------------- factorial(n) invztrans((f+a*z+b*z^2)/(c+d*z+e*z^2),z,n); 2 n 2 n (2*(sqrt( - 4*c*e + d ) + d) *sqrt( - 4*c*e + d )*( - 1) *a*c*e 2 n 2 n - (sqrt( - 4*c*e + d ) + d) *sqrt( - 4*c*e + d )*( - 1) *b*c*d 2 n 2 n - (sqrt( - 4*c*e + d ) + d) *sqrt( - 4*c*e + d )*( - 1) *d*e*f 2 n n 2 + 4*(sqrt( - 4*c*e + d ) + d) *( - 1) *b*c *e 2 n n 2 - (sqrt( - 4*c*e + d ) + d) *( - 1) *b*c*d 2 n n 2 - 4*(sqrt( - 4*c*e + d ) + d) *( - 1) *c*e *f 2 n n 2 + (sqrt( - 4*c*e + d ) + d) *( - 1) *d *e*f 2 n 2 - 2*(sqrt( - 4*c*e + d ) - d) *sqrt( - 4*c*e + d )*a*c*e 2 n 2 + (sqrt( - 4*c*e + d ) - d) *sqrt( - 4*c*e + d )*b*c*d 2 n 2 + (sqrt( - 4*c*e + d ) - d) *sqrt( - 4*c*e + d )*d*e*f 2 n 2 2 n 2 + 4*(sqrt( - 4*c*e + d ) - d) *b*c *e - (sqrt( - 4*c*e + d ) - d) *b*c*d 2 n 2 2 n 2 - 4*(sqrt( - 4*c*e + d ) - d) *c*e *f + (sqrt( - 4*c*e + d ) - d) *d *e*f)/(2 n n 2 *e *2 *c*e*(4*c*e - d )) % Example 1 in Bronstein/Semendjajew, p. 651 f(0):=0; f(0) := 0 f(1):=0; f(1) := 0 f(2):=9; f(2) := 9 f(3):=-2; f(3) := -2 f(4):=23; f(4) := 23 equation:=ztrans(f(n+5)-2*f(n+3)+2*f(n+2)-3*f(n+1)+2*f(n),n,z); 5 3 2 equation := ztrans(f(n),n,z)*z - 2*ztrans(f(n),n,z)*z + 2*ztrans(f(n),n,z)*z 3 2 - 3*ztrans(f(n),n,z)*z + 2*ztrans(f(n),n,z) - 9*z + 2*z - 5*z ztransresult:=solve(equation,ztrans(f(n),n,z)); 2 z*(9*z - 2*z + 5) ztransresult := {ztrans(f(n),n,z)=----------------------------} 5 3 2 z - 2*z + 2*z - 3*z + 2 result:=invztrans(part(first(ztransresult),2),z,n); n n n n n - i *( - 1) + 2*( - 1) *2 - i + 4*n result := ----------------------------------------- 2 % Example 2 in Bronstein/Semendjajew, p. 651 clear(f); operator f; f(0):=0; f(0) := 0 f(1):=1; f(1) := 1 equation:=ztrans(f(n+2)-4*f(n+1)+3*f(n)-1,n,z); 3 2 equation := (ztrans(f(n),n,z)*z - 5*ztrans(f(n),n,z)*z + 7*ztrans(f(n),n,z)*z 2 - 3*ztrans(f(n),n,z) - z )/(z - 1) ztransresult:=solve(equation,ztrans(f(n),n,z)); 2 z ztransresult := {ztrans(f(n),n,z)=---------------------} 3 2 z - 5*z + 7*z - 3 result:=invztrans(part(first(ztransresult),2),z,n); n 3*3 - 2*n - 3 result := ---------------- 4 % Other example: clear(f); operator f; f(0):=1; f(0) := 1 f(1):=1; f(1) := 1 operator tmp; equation:=ztrans((n+1)*f(n+1)-f(n),n,z); 2 equation := - (df(ztrans(f(n),n,z),z)*z + ztrans(f(n),n,z)) equation:=sub(ztrans(f(n),n,z)=tmp(z),equation); 2 equation := - (df(tmp(z),z)*z + tmp(z)) load(odesolve); oderesult:=odesolve(equation,tmp(z),z); 1/z oderesult := {tmp(z)=e *arbconst(1)} preresult:=invztrans(part(first(oderesult),2),z,n); arbconst(1) preresult := -------------- factorial(n) solveresult:= solve({sub(n=0,preresult)=f(0),sub(n=1,preresult)=f(1)},arbconst(1)); solveresult := {arbconst(1)=1} result:=preresult where solveresult; 1 -------------- factorial(n) end; (TIME: ztrans 44920 49280)