Tue Feb 10 12:29:48 2004 run on Linux
*** binomial already defined as operator
*** ~f already defined as operator
% 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
(z - a) *(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
sqrt(z)*cosh(---------) - sinh(---------)*z
sqrt(z) sqrt(z)
---------------------------------------------
2*sqrt(z)
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);
2*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);
2*n
( - 1) *binomial(n,m - 1)
-----------------------------
2*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);
2*n
( - 1) *binomial(n + 3,m - 1)
---------------------------------
2*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 n
a *( - 1)
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);
2*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 2*n
a *( - 1) *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_package odesolve;
*** airy_ai already defined as operator
*** airy_bi already defined as operator
*** ci already defined as operator
*** si already defined as operator
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 for test: 2280 ms, plus GC time: 150 ms